On Projective Planes & Rational Identities

Jason Cornelius Brunson

Thesis submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Sciencein

Mathematics

Dan Farkas, ChairBud Brown

Mark Shimozono

April 28, 2005Blacksburg, Virginia

Keywords: Projective Plane, Intersection Theorem, Desargues’ Theorem,Rational Identity, Storybook

Copyright 2005 by J. C. BrunsonSome rights reserved.

On Projective Planes & Rational Identities

Jason Cornelius Brunson

(ABSTRACT)

One of the marvelous phenomena of coordinate geometry is the equivalence of Desargues’ The-orem to the presence of an underlying division ring in a projective plane. Supplementing thiscorrespondence is the general theory of intersection theorems, which, restricted to desarguian pro-jective planes P, corresponds precisely to the theory of integral rational identities, restricted todivision ringsD. The first chapter of this paper introduces projective planes, develops the concept ofan intersection theorem, and expounds upon the Theorem of Desargues; the discussion culminateswith a proof of the desarguian phenomenon in the second chapter. The third chapter characterizesAutP and introduces the theory of polynomial identities; the fourth chapter expands this discus-sion to rational identities and cements the “dictionary”. The last section describes a measure ofcomplexity for these intersection theorems, and the paper concludes with a curious spawn of thecorrespondence.

Acknowledgments

For structural and moral support during the creation of this document, gratitude is due amultitude of people, whom i hope to thank personally in a more expository manner than this pagewill allow. In particular, however, i must acknowledge my committee, whose contributions haveinfluenced me immeasurably:

• Dr. Farkas, who introduced me to difficult, useful, motivating problems, has asked the ques-tions that have needed answering, and could have cut me so much more slack;

• Dr. Brown, who introduced me to the community of mathematics, continually encouragesthinking outside the classroom, and has helped me keep my head in the clouds;

• Dr. Shimozono, who introduced me to independent discovery, reacquainted me with intuitionand excitement, and made me finally to learn to write.

It gladdens me to think that i am but one of the many students upon whom you all have taken theopportunity to impress.

This document is dedicated to Kelly McLewin, who afforded me both a concrete goal and proofof attainability, and whose camaraderie assured me that i had found my niche.

iii

Contents

1 Some Basics of Projective Geometry 1

1.1 Whence Projective Planes Arise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Putting 3-Space In Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Desargues’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Projective Planes Over Division Rings 12

2.1 Coordinate Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Operations As Projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Projectivities As Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Homogeneous Coordinatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 More About Division Rings 25

3.1 The Projective General Linear Group . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Pappus’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Polynomial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Finite-Dimensional Division Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Rational Identities 40

4.1 Completing the Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Intersection Theorems As Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Storytelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A Coordinatizing the Desarguian Plane 53

B Permuting Perspectors: S3 on R[x] 56

iv

List of Figures

1.1 Quadrangle ABCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Quadrilateral lmno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The Fano Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Some Perspectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Quadrangles Perspective from a Point . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 The Draughtsman of the Lute, Albrecht Durer . . . . . . . . . . . . . . . . . . . . . . 6

1.7 Pentacorn ABCDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.8 Desargues’ Configuration D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.9 Raising Desargues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.10 A Desargues Degeneration δ(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.11 Desargues’ Bigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Origin, Line at Infinity, & Friends . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The Triangle of Reference & A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 The Coordinate Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 The Projectivity ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 The Projectivity τx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 A Harmonic Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 The Projectivity υ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 The Projectivity σx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 The Synthetic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10 Construction of X−1 ·X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.11 Left Multiplicative Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.12 Multiplicative Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.13 (P, l)-Transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Intuitive Structure of P2H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 F5[5X, l0] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

3.3 Additive Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Pappus’ Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Multiplicative Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 The Configuration Iei of Expression ei . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 X · Y and Y ·X on Separate Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 An Open Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 A Word Tree for One Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.1 Collineation τP (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.2 PC[P∞,l∞] and PC[T∞,l∞] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.3 The y-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.4 Coordinatization of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.5 Homology ς(P ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.1 The Original Setup & Projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B.2 Enter A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

vi

List of Tables

3.1 The Quaternion Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

B.1 Permuted Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

vii

1 Some Basics of Projective Geometry

In his 1922 paper [7], geometer Max Dehn discussed some properties of traditional geometrythat are lost in “non-Archimedean” geometries. Specifically, Dehn considered “projective figures”valid (i.e. existing) in certain geometries but not in others, and “point intersection theorems” whichwould hold in these subcategories. As these figures and theorems could be described algebraically,he presented a necessary condition for a relationship not following from Desargues’ and not implyingPappus’ in terms of variables. This paper launched the gradual theory of identities, which strove toclassify all algebraic relationships holding in certain rings but not in others. In this paper we willclassify several such intersection theorems and attempt to unify a somewhat fragmented discussionof the algebraic correspondences in the literature.

1.1 Whence Projective Planes Arise

Let us place ourselves in the familiar euclidean plane; we make certain observations, as postulatedby Euclid himself in his Elements:

Though our view be one-dimensional, we observe both lengthless objects (points) andlengthy objects such as line segments, circles, and even entire lines (which can be arbi-trarily long). We also notice that any two distinct points can be connected by a straightsegment, and that this segment lies on a line, and that only one line can be got this way.

Several pairs of lines intersect, and when they do it is at a point, also singular in thisrespect. However, some lines do not intersect. When a third line, a transversal, intersectsthem, they form identical pairs of interior angles; however far along one we travel, theother remains a fixed distance away.

These are parallel lines, whose behavior was the last feature of the plane postulated by Euclid.Since parallelism is clearly an equivalence relation these lines come in equivalence classes, calledpencils. We accept their existence, and we are happy, for these euclidean axioms are consistent.

Now we, being three-dimensional thinkers, can rise above this world, and look down uponit: There are those two points, with the line through both . . . and over there are two linescut by a transversal! Not too far away we can see where they meet.

But wait—see those two parallel lines? (We know because of the interior angles theymake with that transversal.) They’re always the same distance apart, but the fartherthey get from us, the closer they seem to each other. We might even declare, “theyintersect at a vanishing point on the horizon”.

In fact, if we were to declare this for every pair of parallel lines then we would get ahorizon full of vanishing points, one for each pencil. We could even declare the horizonto be a line containing all these vanishing points, and no others. Then we’d have a worldin which every two points share a line and every two lines share a point! One mightalmost think points and lines would be interchangeable. . . .

The euclidean plane is an example of an affine plane, a structure of points, lines, and incidencesbetween them satisfying axioms similar to but looser than Euclid’s. Some examples of affine planes

1

include R2, the xy-plane, which is typically identified with the euclidean plane, in which pointsare given by ordered pairs and lines by linear equations, the usual way; and such finite planes asZ3 × Z3, in which points are still given by ordered pairs but lines are given by order-3 subgroupsand their cosets. But, looking at incidence, this second plane is identical to F3

2, with lines by linearequations; in fact, if F is any field then F 2 can be made into a perfectly valid affine plane by simplydefining lines by linear equations.

What of the horizon, and its vanishing points? It turns out that, if A is a affine plane, we canappend one new point for each pencil of parallel lines and a new line containing precisely thesepoints to obtain a curious structure called the projective completion of A , denoted A . We thencall the appended points “points at infinity” and the appended line the “line at infinity”.

We call this a completion, but what has been completed? We now introduce a fundamentalconcept, the palette of this paper.

Definition. A projective plane is an incidence structure consisting of a set of points A, a setof lines L, and a set of incidences I ⊂ A× L subject to the following conditions:1

(i) any two distinct points determine a unique line incident with both (their join);

(ii) any two distinct lines determine a unique point incident with both (their intersection); and

(iii) there exists a quadrangle.

67%

C

B

A

D

Figure 1.1: Quadrangle ABCD

A “quadrangle” consists of four points in A, no three of which are collinear, generally togetherwith their pairwise joins. A quadrangle thus includes

(42

)= 6 lines; otherwise the quadrangle would

“degenerate” to a “triangle” or single line. (Sometimes the configuration of Figure 1.1 is called a“nondegenerate quadrangle” to emphasize this distinction.) In the future we will also refer to setsof four triplewise-noncollinear points—also called “admissible points”—as quadrangles, sometimeswith select lines included.

It can also be seen easily that Axiom (iii) is logically equivalent to the pair of alternatives

(iii)a there exist three noncollinear points and

(iii)b every line contains at least three points.

In general, any set of points and set of lines with some set of incidences between them comprisea configuration. The set of incidences is taken to be exclusive, meaning that a point A and a line lwith (A, l) not in the set of incidences are considered nonincident (as opposed to “maybe incident”).

1We will equate (A, l) ∈ I with familiar phrases like “A lies on l”, “l passes through A”, and “A ∈ l”.

2

A triangle is then a configuration of three points, three lines, and six incidences, while a quadrangleis made up of four points, six lines, and 12 incidences. The diagonal points of quadrangle ABCDare the three intersections (AB)(CD), (AC)(BD), and (AD)(BC) of pairs of opposite sides.

n

mo

l

Figure 1.2: Quadrilateral lmno

A configuration of four lines, no three of which are concurrent (incident with a common point),and their six points of intersection is called a quadrilateral ; the joins of opposite corners are calleddiagonal lines. There is a bijection between the points and lines of a quadrilateral and the linesand points (respectively) of a quadrangle; we call the quadrilateral the dual of the quadrangle. Thedual of a triangle is then another triangle, an equivalent configuration; such a configuration is calledself-dual.

Given any two points A,B or two lines l,m in a projective plane, we postulate to constructtheir join or intersection in the plane. We will write AB for the join of A and B and lm for theintersection of l and m.

Observe that no three of the lines l = AB, m = BC, n = CD, o = DA above can be con-current, so the configuration consisting of these four lines and their points of intersection is itselfa quadrilateral. (Compare Figures 1.1 and 1.2.) Thus the existence of the quadrangle implies theexistence of its dual. This observation reveals that if P = (A,L, I) is a projective plane then so isits dual P∗ = (L,A, I). (Check out the duality between Axioms (i) and (ii).) This principle ofduality is key, for (among other attributes) it says that any consequence of these axioms yields asecond, equally valid dual consequence by interchanging the words “point” and “line” and slightlyrewording the language of incidence.

Example (the Fano Plane). Consider the configuration below, a quadrangle ABCX whose diago-nal points M , N , and O are collinear (or, if you prefer, a quadrilateral abcx whose diagonal lines m,n, and o are concurrent; or perhaps a quadrangle AOXN , or a quadrilateral conb, or . . . ). Taking

x

c

b

a

n

mo

O M

N

X

CA

B

Figure 1.3: The Fano Plane

3

A and L to consist of the seven points and seven lines pictured, respectively, under the incidencesdepicted this configuration is a projective plane, called the Fano Plane. This is the “smallest”possible projective plane (by point or line count), and is the projective completion of the smallestaffine plane F2

2 of four points and six lines. Observing closely the chosen labeling scheme, we canalso see that the Fano Plane is self-dual!

Example (antipodal points). Let S be a sphere in euclidean space, the set of points a fixed distancer from some center O. Designate the great circles C to be “lines” and pairs of antipodal points tobe “points”. The axioms for a projective plane are then handily satisfied.

Any affine plane can be completed; conversely, any projective plane can be “decompleted”: Takeany line l ∈ L and call it the “line at infinity”; call its points “points at infinity”. Declare the pencilof lines through each point at infinity to be parallel. Then remove the line at infinity and its points.The resulting configuration is an affine plane, as can be checked axiomatically.

1.2 Projectivities

It is only natural to describe “automorphisms” in a projective plane; while we need not findthem all, a few will be of importance in the pages to come. First, a bit of notation. Expanding upona previous definition, we call the collection of lines incident with a fixed point A the pencil of linesthrough A, written [A]. Dually, the range of points on a line l is written [l].

Take a line l and a point A off l. Then the map [A] → [l] sending each line through A to itsintersection with l is one-to-one and onto. (Stationing ourselves at viewpoint A, we are using l asa canvas upon which to project our lines of sight.) If we designate another point B off l we canbiject these points on l with the lines through B the same way; we then have a composite mapψ : [A] → [B], called a perspectivity. The line l from which A and B are perspective is called theperspectrix. If x′ = (xl)B; that is, if ψ(x) = x′, then we write the correspondence x l

[x′. (Notice

x

lx'

m

l

x

X'

A

B

A

X

X

Figure 1.4: Some Perspectivities

that this is equivalent to saying that lines x, l, x′ are concurrent.) Dually, a map ϕ : [l] → [m] onranges of points can be defined using a point A off l and m, namely ϕ(X) = (XA)m; this is alsocalled a perspectivity, and written X A

[X′ when X ′ = ϕ(X). A is called the perspector.

Two triangles ABC and DEF are perspective from a point O if their vertices can be paired upin such a way that each pair lies on a line through O; we then write ABC O

[DEF . The same goes forquadrangles. For more complicated configurations we must require that the mapping that switchespaired points is an “isomorphism”, but that term will come later, and we needn’t worry aboutperspective configurations larger than these. Dually, triangles and quadrilaterals are perspectivefrom a line if their lines can be paired up similarly.

4

Figure 1.5: Quadrangles Perspective from a Point

A composition

x1l1

[x2l2

[ · · ·ln−1

[ xn

of finitely many perspectivities is called a projectivity; since every perspectivity is bijective, theprojectivity is bijective. The composition of two projectivities is naturally also a projectivity. If welabel a projectivity φ we write x1

φ

Zxn, or just x1Zxn when the map is understood from context.Likewise, a projectivity

X1A1

[ X2A2

[ · · ·An−1

[ Xn

which maps [l] to itself is a range bijection of l, and we can write X1ZXn.Every projectivity φ : [A] → [A], as a composition of perspectivities ϕ1◦· · ·◦ϕn−1, has an inverse

projectivity φ−1 = ϕn−1 ◦ · · · ◦ ϕ1 that gives φ ◦ φ−1 = φ−1 ◦ φ = id[A]. So the projectivities from apencil of lines through a point A to itself form a group, called PJ(A), a subgroup of the symmetricgroup of all permutations of the lines through A. Dually we call the group of projectivities fromthe range of line l to itself PJ(l).

Much more can be said of perspectivities and projectivities, but we will have little use for morethan what we have; the reader is referred to [8] and [6] for further reading. Meanwhile, we shallbroaden our horizons—by one dimension.

1.3 Putting 3-Space In Perspective

The origins of perspective as an artistic technique lie somewhat before its recorded historybegins, between the Optica of Euclid (some 300 years before the gregorian calendar begins) and thefirst public demonstration around 1413 by Filippo Brunelleschi. Brunelleschi had applied an arsenalof mirrors to the laws of perspective to paint and display an image (the church of San Giovanni diFirenze in Florence) which was sight-accurate—a viewer with zer eye at a specific location wouldsee precisely the same image (up to painting imperfections) by viewing the painting as by viewingthe church itself. (Kubovy exhibits this method in glorious detail in Ch. III of [9].)

More than a century later the renowned artist (and ardent mathematics student) Albrecht Durerlaid out, for the first time in wide circulation, the theory and practice of perspective in his Treatiseon proportion. His is a championed Renaissance work, and etchings exist of several of his drawingdevices. In the etching of Figure 1.6 can be seen the basic elements of the perspective technique:space (represented by the three-dimensional lute), canvas (the vertical panel of the device), andviewpoint (the hook on the wall threaded by the weighted string). To anyone with zer eye at theviewpoint, the painted lute would match up with the actual lute (via opening and closing the panel)precisely.

5

Figure 1.6: The Draughtsman of the Lute, Albrecht Durer

Were the etching to depict an infinitely long table, we could say that the plane of the table meetsthe plane of the sky at the horizon, or line at infinity. So far in this paper we have no language withwhich to discuss more than one plane; but the stage for perspective in art is set in 3-dimensionalspace, and it is in this spirit that we consider a new definition. For the sakes of simplicity andfamiliarity we shall think in terms of containment, rather than incidence.

Definition. Let A be a set of points, L a set of subsets of A called lines, and E a set of subsetsof A called planes. Then (A,L, E) is a projective 3-space when

(i) any two distinct points determine a unique line through both,

(ii) any three noncollinear points determine a unique plane containing them all,

(iii) any plane and any line not contained therein determine a unique point on both,

(iv) any two distinct planes determine a unique line contained in both, and

(v) there exists a pentacorn.

The mythical “pentacorn” is a collection of five points, no four of which are coplanar, togetherwith the ten lines and ten planes they determine. Analogously to the planar case, its existence isequivalent to the pair of axioms that

(v)a there exist four noncoplanar points and

(v)b every line contains at least three points.

6

E

A

D

C

B

Figure 1.7: Pentacorn ABCDE

An immediate consequence of Axioms (ii) and (i) is that a line and a point off the line determinea unique plane. Going further, consider two lines l and m in the same plane Π. We can thendetermine plane P containing both l and some point A off Π. P and m then share exactly one pointB, but since Π∩P = l, we see that B is also on l. Thus l and m share a unique point. We concludethat every plane in a projective 3-space is itself a projective plane!

We can also obtain the projective completion of an affine 3-space (e.g. euclidean space or F 3 forsome field F ) analogously to the completion of the plane: For each line define a point at infinity(vanishing point) also on the line; for each plane define a line at infinity (horizon line) also in theplane, containing the vanishing points of all its lines; and define a plane at infinity (the “celestialplane”) containing all the horizons.

We can actually prove something curious about the projective planes that lie in a projective3-space, using a technique much like Durer’s. To begin, we obtain the following result from theaxioms:

Theorem 1.1. In a projective 3-space, if two triangles are perspective from a point then they areperspective from a line.

Q

P

R

O

AB

C

A'

B'C'

Figure 1.8: Desargues’ Configuration D

Proof. If noncoplanar triangles ABC and A′B′C ′ are perspective from O (respectively) then eachof the intersections AB∩A′B′, BC∩B′C ′, and CA∩C ′A′ lies on ABC∩A′B′C ′, which is a line. Forthe coplanar case, the configuration can be “reverse-durered” into 3-space: Designate a viewpointV (in Figure 1.8, for instance, yours) outside the plane containing the configuration. Let O1 be a

7

A 1

A A'

O1

V

O

Figure 1.9: Raising Desargues

third point on the line V O. Note that V , O1, A, and A′ are coplanar; find A1 = V A ∩ O1A′. By

identical constructions we can get the triangle A1B1C1 which is definitely not coplanar with A′B′C ′

but is also definitely perspective to it from point O1. By the result for noncoplanar triangles theintersections P1, Q1, and R1 must be collinear. But, for instance, plane O1B

′C ′ intersects line BCin exactly one point—P . So (P1, Q1, R1) = (P,Q,R) and P , Q, and R are collinear.

1.4 Desargues’ Theorem

The statement of Theorem 1.1 is also true in the euclidean projective plane; as a property ofthe plane we will refer to it as Desargues’ Theorem. However, the property cannot be derivedfrom the axioms of page 2; in fact, it isn’t true in all projective planes (which we shall see bycounterexample at the end of this section). It is extremely useful to have around, however, and isnot especially sectarian. For instance, Theorem 1.1 implies that every plane of a projective 3-spaceis “desarguian”, so every projective plane that can be embedded in some projective 3-space is alsodesarguian. The configuration of Figure 1.8 is also called Desargues’, and we shall label it D .

The reason we call this property a “theorem” is that it is of a type, satisfied by some projectiveplanes but not others, called an intersection theorem. We now restrict to the world of projectiveplanes and lead up to a rigorous definition of this concept.

Definition. Take C1 = (A1,L1, I1) and C2 = (A2,L2, I2) to be two configurations. A (configura-tion) morphism from C1 to C2 is a map

ψ : C1 → C2

which takes points to points and lines to lines in such a way that if (A, l) ∈ I1 then (ψ(A), ψ(l)) ∈I2—that is, ψ preserves incidence. If ψ is bijective, i.e. one-to-one, then it is an isomorphism;otherwise it is degenerate.

For instance, take D . Choose any point, and examine the other points on the lines through it(for example, the three lines through B′ can be written OB, A′R, and C ′P ). These pairs comprisetwo triangles perspective from the chosen point (OA′C ′ and BRP are perspective from B′). Thesetriangles are also perspective from the line through the remaining three points (A, C, and Q)!Thus any point of D may serve the role of O, either of the two triangles perspective from it mayserve as the “first” ABC (the other as A′B′C ′), and the three points of this triangle may belabeled in any order; so there are 10 · 2 · 3! = 5! configuration isomorphisms from D to itself, aptlycalled automorphisms. (In fact, AutD ,the group of automorphisms of Desargues’ Configuration, isisomorphic to S5, the symmetric group on five letters—for instance, the five points of the pentacorn.

8

This follows from the observation that these two configurations are duals in the sense that there is abijection from the points of one to the planes of the other. Since Axiom (iii) basically says that anythree noncollinear planes determine a unique point, we see that when (A,L, E) is a projective 3-space, so is (E ,L,A), and it is this spatial principle of duality which restricts to the configurations.)

Q

P

RO

A

BC

Figure 1.10: A Desargues Degeneration δ(D)

On the other hand, there are also many degenerate morphisms from D to itself, one examplebeing the map δ which fixes O,A,B,C, P,Q,R, sends A′, B′, C ′ to A,B,C, and maps lines accordingto incidence. In effect, δ “degenerates” the configuration D into the simpler configuration δ(D)(Figure 1.10) of only seven points, seven lines, and 18 incidences. These morphisms will be importantto consider in the definitions and discussions to follow.

Definition. An intersection theorem is a configuration C = (A0,L0, I0) (finite, for our pur-poses) together with an incidence relation (A, l) (where A and l are in C ). We will write thistheorem C [A, l], to be read “C unites A and l”.

Suppose P is a projective plane with incidences I. For an isomorphism ψ : C → C ′ ⊂ P, theintersection theorem is said to hold under ψ if (ψ(A), ψ(l)) ∈ I—if the point and line are indeed“united”. If for every configuration isomorphism ϕ : C → P the incidence (ϕ(A), ϕ(l)) is in I,then C [A, l] is called a universal intersection theorem of P, and P is said to satisfy C [A, l].

Cϕ↓C ′ ⊂ P

We can now write Desargues’ Theorem D [P,QR], inferring from the notation that the incidence(P,QR) is not specified in D . To say that P is desarguian is to say that D [P,QR] is a universalintersection theorem of P. The collection of intersection theorems that hold in a projective planeprovides a great amount of information about the plane itself, as we shall see. From here on, unlessexpressly stated, all intersection theorems are assumed to be universal. To avoid confusion withproved results, they will not be numbered.

Aside. If we label the lines in D creatively, we can give D (the hypothesis of Desargues’ Theorem)the form D = (A0,L0, I0) with

A0 = {O,A,B,C,A′, B′, C ′, P,Q,R}L0 = {o, a, b, c, a′, b′, c′, p, q, r}I0 = {(O, p), (O, q), (O, r), . . . , (P, o), (Q, o)}.

The names given to the lines reflect the points’ names such that the incidences of D [R, o] canbe depicted in the “desargues bipartite graph” of Figure 1.11. In this graph, points and lines are

9

c

a

R

P c'

a'C'

A'

r

p

C

A

o

Q

b'

b

B'

B

q

O

Figure 1.11: Desargues’ Bigraph

represented by black and white vertices, respectively, and incidences determine the edges. Anyconfiguration in a projective plane can be viewed as a bipartite graph (bigraph) in this way. Itmay be clearer from the graph that any incidence may be substituted for (R, o) to obtain anequivalent statement, just as there is a graph isomorphism from (R, o) to any other edge (witheither orientation, in fact; this graph makes very clear that Desargues’ Configuration is self-dual:just switch lowercase vertices with capital vertices). C

We now affirm that Desargues’ Theorem does not follow from the axioms.

Example (a nondesarguian projective plane). Let D be a desargues configuration and generatethe free projective plane over D by the following iterative steps:

• Start with P0 = D .

• If n is even: For every pair of lines l,m ∈ Pn, if no point is incident with both l and m thenappend the point lm incident only with l and m. Call the new configuration Pn+1.

• If n is odd: For every pair of points A,B ∈ Pn, if no line is incident with both A and B thenappend the line AB incident only with A and B. Call the new configuration Pn+1.

• Define P(D) =⋃∞n=0 Pn, the free projective plane over D .

P(D) is indeed a projective plane; every two points (which must be in some Pn) share a line(in Pn+1), every two lines share a point (similarly), and D itself contains four “quadrangular”(triplewise noncollinear) points.

Now suppose (contradictorily) that Desargues’ Theorem holds in P. Take an arbitrary pointV outside the original D and find two triangles perspective from V . Then there is a configurationD ′ isomorphic to D but not contained in P0. Each object appears first in some Pn, so pick thelargest such n—that is, the smallest integer such that D ′ ⊂ Pn. For the moment, suppose n isodd, meaning some point A appended at the nth step completes the configuration. This means,though, that in Pn the point A is incident with only two lines (the two that determine it). Sinceevery object in D is incident with three others, we have a contradiction. A dual argument for evenn (with some line l appended at the nth step) completes this check that P(D) is nondesarguian.

This argument works for any free projective plane and any intersection theorem which (likeDesargues’) is confined, in the sense that every object is incident with at least 3 others. We think

10

of a confined configuration as the heart of an intersection theorem; for instance, for a configurationC = (A0,L0, I0) including an object (say a point P ) denote

C r {m} = (A0,L0 r {m}, I0 r {(∗,m)}).

If, in the intersection theorem C [A, l], a line m distinct from l is incident with only two points,then (C r {m})[A, l] is an equivalent theorem—that is, each implies the other—since Axiom (i)implies the existence of m from C r {m}. (Observe that P(Q), the free projective plane over aquadrangle Q, contains no confined configurations.)

It is not always true that two intersection theorems are equivalent when one can be “confined”to the other. For instance, if we take F to be the Fano Plane missing one incidence, say (M,NO),then F [M,AB] is obviously not an intersection theorem in the Fano Plane; but (F ∪{P})[M,AB]is an intersection theorem by default—the 10-point configuration F ∪ {P} doesn’t exist in theplane of 9 points. However, we can see that this is strictly a quirk of finite projective planes.

Let C be a finite configuration with O the set of objects in C of fewer than 3 incidences. LetC ′ = C r O, also a configuration, with O′ the set of objects in C ′ of fewer than 3 incidences.Continue with C (i+1) = C (i) r O(i) until the first C (k) is confined. Define 〈C 〉 = C (k) to be theconfinement of C . For any intersection theorem C [A, l], perform the same withdrawals to get〈C [A, l]〉 with the condition that A and l are never removed. (Be careful—if we take D to be without(R, o), as when Desargues’ Theorem is written D [R, o], then 〈D [R, o]〉 = D [R, o] while 〈D〉 = ∅,the “empty configuration”!)

Lemma 1.2. Let C [A, l] be a finite intersection theorem and P an infinite projective plane, thenP satisfies C [A, l] if and only if it satisfies 〈C [A, l]〉.

Proof. By duality and finiteness we need only consider the removal of one point P 6= A, then applyinduction on the number of objects in C . Let C ′ = C r {P} and consider the following cases in thecontext of P, an infinite projective plane.

If P is “stray”, i.e. incident with no lines, then since any map fixing C ′, sending P to P ′ /∈ C ′

is an isomorphism, C [A, l] ⇔ C ′[A, l] provided there exists some P ∗ ∈ P r C ′. Since C ′ finite, thisis always the case.

If P is only incident with line m in C , C [A, l] ⇔ C ′[A, l] provided there is always a point Pon m other than those in C ′; again, since infinite projective planes possess infinite lines, this isguaranteed.

If P is incident with exactly two lines m and n, i.e. P = mn, then the intersection mn is not inC ′ but is guaranteed to be in P by the axioms, as before.

Note. Theorems of another type crop up also. Fano’s Theorem states that the three diagonalpoints of a complete quadrangle are never collinear. That a projective plane P satisfies Fano’sTheorem is just the affirmation that the Fano Plane is not a subconfiguration of P. Such “non-intersection theorems” also provide information about the plane, but of a different sort, which wewill see later.

11

2 Projective Planes Over Division Rings

We have seen a few projective completions of F 2 for fields F such as F3 (played by Z3) and R.It is time we generalize this concept.

2.1 Coordinate Homogenization

Let F be a field, and consider the vector space V = F 3. Let A be the set of 1-dimensionalsubspaces and L the set of 2-dimensional subspaces—lines and planes, respectively, through theorigin (0, 0, 0). Any A ∈ A then has the form

A = {λ~v | λ ∈ F}

for some nonzero vector ~v, while the l ∈ L can be defined by linear equations of the form

l : ax+ by + cz = 0

for coefficients a, b, c, not all zero. For any pair of such subspaces A and l, check that either A∩l = Aor A ∩ l = {~0}; we may then adopt the convention that A and l are “incident” when A ⊂ l as sets.Since every field contains 0 and 1, each vector space F 3 contains the four dimension-1 subspacesλ(1, 0, 0), λ(0, 1, 0), λ(0, 0, 1), and λ(1, 1, 1), any three of which span V , so no three of which lie ina common dimension-2 subspace. Referring to dimension-1 subspaces as “points” and dimension-2subspaces as “lines”, we have then satisfied the quadrangle axiom, making (A ,L ,⊂) a projectiveplane (as the other axioms are trivially satisfied).

Meet P2F , the projective plane over F .One feature of P2F is that as a projective plane it is identical—via a bijection of points that

preserves lines—to F 2, the projective completion of the affine plane F 2. To see this (proof isomitted), consider the plane Π determined by z = 1 and make the correspondence

A↔ Π ∩A

between points in Π and “points” of P2F . Lines then correspond via

l : ax+ by + cz = 0 ↔ Π ∩ l : ax+ by = −c, z = 1,

which are clearly lines in Π in the usual sense. Thus

Π = F 2 × {1} ↔ F 2

in the natural way, and a bit more insight will reveal that the “line at infinity” of F 2 correspondsto the “line” z = 0 of P2F .

While the plane z = 1 serves as a canvas for the affine component of P2F , any sphere about theorigin will serve as a complete canvas for the projective plane, with antipodal points representingthe same “point” of P2F . Hence the two basic models are unified by this definition.

To discuss these subspaces more concisely, we introduce some new notation. Write each vector ~vin component form ~v = (x, y, z). Define an equivalence relation ∼ on V ×, the set of nonzero vectors

12

l!

l

A

A 0

Figure 2.1: Origin, Line at Infinity, & Friends

in V = F 3, by whether the vectors describe the same point in P2F . Then (x1, y1, z1) ∼ (x2, y2, z2)when there is some λ ∈ F such that (λx1, λy1, λz1) = (x2, y2, z2). Write (x :y :z) for the equivalenceclass containing (x, y, z). These are homogeneous coordinates, and we will use them to identifypoints in the projective plane.

Dually, the line defined by the planar equation ax + by + cz may be assigned homogeneouscoefficients [a : b : c] subject to the same equivalence relation. A point (x :y :z) and line [a : b : c] arethen incident when their scalar product is zero. We may now specify the origin A0 = (0 :0 :1) andthe line at infinity l∞ = [0:0 :1].1

We can generalize yet further.

Definition. Recall that if R is a ring then the units of R are those elements with “inverses”—elements r ∈ R for which there exist s ∈ R such that rs = sr = 1. A division ring is a ring inwhich every nonzero element is a unit.2 The inverse of r ∈ D is written r−1.

The nonzero elements of D then form a group D×. The commutative division rings are preciselythe fields; noncommutative division rings are called skew fields.

Let D be a division ring, and let V = D3. In V ×, define the equivalence relation ∼ on vectors by(x1, y1, z1) ∼ (x2, y2, z2) when there is some (nonzero) λ ∈ D such that (x1λ, y1λ, z1λ) ∼ (x2, y2, z2).(Notice the right multiplication, a significant choice when multiplication may not be commutative.)Let (x :y :z) denote the equivalence class of (x, y, z), and let A denote the set of equivalence classes.We have our “points”.

For any vector (a, b, c) ∈ V ×, consider the set of all vectors (x, y, z) that satisfy ax+by+cz = 0,i.e. that are right-orthogonal to (a, b, c). Since we are now dealing with left multiplication, therelation ∼ preserves right-orthogonality: for any λ ∈ D×,

ax+ by + cz = 0 ⇔ axλ+ bxλ+ czλ = 0.

Let [a :b :c] denote the set of classes right-orthogonal to (a, b, c), and let

L = {[a :b :c] | (a, b, c) ∈ V ×}.

We have our “lines”. This time, however, we have [a1 :b1 :c1] = [a2 :b2 :c2] when there is some λ ∈ D×

such that (λa1, λb1, λc1) = (a2, b2, c2); points and lines are not subject to the same equivalence1Thus, expanding to the terminology of 3-space, “the eye is the dual of the heavens”.2The reader is alerted that the term “division ring” is defined a variety of ways depending on convenience, context,

or author preference; for all of these reasons we will stick to this definition throughout this paper.

13

relation anymore! They do, however, still form a projective plane P2D, the projective planeover D.

We may also construct a projective 3-space from D4 in similar fashion. Points are assignedhomogeneous coordinates (x : y : z :w) and planes homogeneous coefficients [a : b : c : d], with linesdefines as the intersections of planes.The projective plane P2D then appears as the plane [0 :0 :0 :1]in P3D, the set of points (x :y :z :0). Theorem 1.1 then implies that Desargues’ Theorem holds inP2D.

2.2 Operations As Projectivities

What does the geometry of the projective plane P2D say about D? We examine the desarguesproperty and its consequences. Let us name the origin A0 = (0 : 0 : 1), the line at infinity l∞ = [0 :0 :1], and the x- and y-axes l1 = [0:1 :0] and l0 = [1:0 :0]. We also then have

A∞ = l1l∞ = [0:1 :0][0 :0 :1] = (1:0 :0)

andA′∞ = l0l∞ = [1:0 :0][0 :0 :1] = (0:1 :0),

which complete the triangle of reference. This specific configuration will indeed serve a referential

A *

(0:1:0)

(0:0:1)(1:0:0)

(1:0:1)

Figure 2.2: The Triangle of Reference & A1

role for the remainder of the paper. Finally, let us identify A∗ = (1:1 :1), a point not on any side ofthe triangle of reference, and construct A1 = (A′

∞A∗)l1 = (1 : 0 : 1), a point on l1 “one unit” fromA0.

Remember that the points of l1 are A∞ plus those of the form (x : 0 : 1), one for every x ∈ D.We should therefore be able to express arithmetic operations in D as mappings of l1 to itself. Weknow something about such mappings—we have a group of them, the projectivities PJ(l1)!

Allow us a modest setup, adapted from Rowen [10]: Let A = (a : b : 1) be an arbitrary point offl1 and l∞. Define

l = A∞A = (1:0 :0) (a :b :1) = [0:−b−1 :1]l′ = A0A = (0:0 :1) (a :b :1) = [−a−1 :b−1 :0]l′′ = A1A = (1:0 :1) (a :b :1) = [(1− a)−1 :b−1 :1]

so thatI = l′l∞ = [−a−1 :b−1 :0] [0 :0 :1] = (a :b :0)I ′ = l′′l∞ = [−1:(a− 1)b−1 :1] [0 :0 :1] = (a− 1:b :0),

14

by solving the equations. (In the affine plane, we would call l1 and l parallel.)

l

l''

l'

l!

l 0

l 1A 1

A'!

A!

A 0

I

I'

A

Figure 2.3: The Coordinate Setup

While it is easy to check that the joins and intersections above are valid, to find the join orintersection of two arbitrary points or lines involves a messy bit of algebra, which will come intoplay later. For now, let’s define some projectivities from l1 to itself.

Projectivity 1. Our first perspectivity will have perspector A and take l1 to l∞. If X = (x :0 :1)

–X

X'

X!

X

Figure 2.4: The Projectivity ρ

is any other point on l1 thenX

A

[X∞,

where X∞ = (a− x :b :0). Next take l∞ to l via perspector A0. This time

X∞A0

[ X′

with X ′ = (a− x :b :1). Finally, via I we send l back to l1, so that

X ′ I

[X∗

with X∗ = (−x :0 :1). Thus we have interpreted negation in D as the projectivity

ρ : [l1]A

[ [l∞]A0

[ [l] I

[ [l1]

15

in PJ(l1). For convenience we will denote by −X the image of a point X on l1 under ρ. (Checkthat, indeed, A0

ρ

ZA0—that is, 0 is its own negative.)

Projectivity 2. Remember the point X∞ from the previous construction, and build the projec-tivity

τx : [l1]I

[ [l]X∞[ [l1].

We let Y = (y : 0 : 1) be another arbitrary point on l1 other than A∞. In Figure 2.5, Y0 = (Y I)l =(a+ y :b :1), so τx(Y ) = (X∞Y0)l1 = (x+ y :0 :1). We have conquered addition in D, and shall givethis point the deserving name X + Y .

X+Y

Y 0

X!

XY

Figure 2.5: The Projectivity τx

Aside. The construction of X +X, which we will call 2X, has an interesting feature: Let QX bethe quadrangle determined by the points A, X∞, I, and X1 (darkened in the picture). Two of thediagonal points of QX lie on l1: A∞ and X. The other two diagonals intersect l1 in A0 and 2X.When four collinear points have this property, they are called a harmonic quadruple, and we writeH(A∞X,A02X). Notice also that H(A∞kX, (k − 1)X(k + 1)X) for each positive integer k, as thequadrangles shift along l∞. A sequence of points like {Xi} with this property is appropriately calleda harmonic sequence.

4X!

3X!

2X!

X 0

5X 4X 3X 2X

X!

X

Figure 2.6: A Harmonic Sequence

16

Different quadrangles are formed when we use kX1 instead of kX∞ to construct the (k + 1)X,and remember that A is an arbitrary point; the fourth point in a harmonic quadruple is uniquelydetermined by the other three. C

In Projectivities 1 and 2 the two perspectors A and A′1 were collinear with A∞ = l1l∞. In the

next perspectivity, this will not be the case, and we’ll see how the difference manifests.

Projectivity 3. Defineυ : [l1]

A

[ [l∞]A1

[ [l′] I′

[ [l1],

a perspectivity similar to ρ. If we follow an arbitrary point X = (x :0 :1) on l1 and off l∞, we find

XA

[X∞A1

[ X′′ I′

[υ(X),

where we recognize X∞ from before, X ′′ = (a : b :x), and υ(X) = (x−1 : 0 : 1). Important to noticehere is that υ sends A0 to A∞, which does not correspond to a member of D. This is appropriate,as 0−1 has no meaning in D; let us merely define υ on the points of l1 other than A∞ and A0. Likebefore, let’s denote υ(X) = X−1.

X –1

X''

X!

X

Figure 2.7: The Projectivity υ

Projectivity 4. Again keep track of X∞ from before. Define

σx : [l1]I′

[ [l′]X∞[ [l1],

building off υ in the same manner as τx built off ρ. We take our other arbitrary point Y throughthe motions to obtain Y1 = (Y I ′)l′ = (a : b : y−1) and σx(Y ) = (X∞Y1)l1 = (xy : 0 : 1). Finally, wename this point X · Y to avoid confusion with the join XY = l1 of X and Y .

We now have a way, given any starting points Xi on l1, to find the point on l1 corresponding toany member in D generated by the xi using the operations −, −1, +, and ·, when these operationsare defined.3

3In fact, we have more than that: the reader may have noticed that, according to these constructions, A0−1 = A∞,

and almost all arithmetic can be done with A∞ in place of some (x : 0 : 1); for instance, X · A∞ = (X∞A∞1)l1 =(X∞I)l1 = l∞l1 = A∞. However, this construction is undefined when X = A0, since X∞A∞1 = II is not a well-defined line. Excepting this construction and its companions A∞ ·A0 and A∞ + A∞, the point A∞ acts as we wouldexpect an infinite value to act. For reasons that will become clearer later on, we omit these exceptional constructions,focusing on those which correspond to meaningful calculations in D, and leaving A−1

0 undefined.

17

X·Y

Y 1

X!

X

Y

Figure 2.8: The Projectivity σx

2.3 Projectivities As Operations

Let us step back for a moment. We’ve found ways to identify sums, products, and inverseson l1 using the triangle of reference as a framework, the fingers that define our cat’s cradles ofprojectivities. Yet it is everywhere in the structure of the projective plane P2D that the essenceof skew-fieldness lies; can we extract it without clinging so to its coordinates?

As an experiment, let P be any desarguian projective plane and let ABCD be any quadranglein P. Make the assignments A0 = A, A∞ = B, A′

∞ = C, and A∗ = D. (We shall call this thequadrangle of reference by virtue of its applicability, to follow.) Construct l1, l∞, and A1 as before.Choose some arbitrary A off this quadrangle and again ascertain l, l′, l′′, I, and I ′. We have a

l 0

l''

A 1

A'!

ll'

l 1

l!

I'

I

A

A!

A 0

Figure 2.9: The Synthetic Setup

familiar setup and several projectivities in PJ(l1) at our disposal, so let us see how they act—andinteract—on [l1].

Let ∆ = l1 r {A∞}. Where P is any point in ∆, and with P∞A

[P where applicable, define the

18

projectivities

ρ : [l1]A

[ [l∞]A0

[ [l] I

[ [l1]

τP : [l1]I

[ [l]P∞[ [l1]

υ : [l1]A

[ [l∞]A1

[ [l′] I′

[ [l1]

σP : [l1]I′

[ [l′]P∞[ [l1]

as before. Then, for every P,Q ∈ ∆ except P = A0 on the third line, define the operations

−P = ρ(P )P +Q = τP (Q)P−1 = υ(P )P ·Q = σP (Q),

which are well-defined in ∆. What, then, is ∆ under these operations?

In Figure 2.7, we have X ′′ I′

[Y−1; in the notation of Figure 2.8, we would say X ′′ = (X−1)1

(subtly applying Axiom (ii)). This observation confirms that

X ·X−1 = (X∞X−1

1)l1 = (X∞X′′)l1 = A1.

Similarly, we can get X + (−X) = A0, and it is just as easy to show A1 · X = X · A1 = X andA0+X = X+A0 = 0. Using familiar language to describe the operations above, we may now affirmthat ∆ has well-defined additive and multiplicative identities and right-inverses. Is ∆ a divisionring under + and ·?

X –1!

X 1

X –1

X''

X!

X

Figure 2.10: Construction of X−1 ·X

While in Figure 2.10 X−1 · X appears to be the same point as A1, we cannot be immediatelysure that this is so. In order to show that it is so, we need to prove the intersection theorem thatthe line X−1

∞X1 is incident with the point A1. As a configuration, the theorem reads

A = {A1, A, I′, X,X∞, X

′′, X−1, X−1∞, X1}

L = {l1, l∞, l′, l′′, XA,A1X∞, I′X ′′, X−1A,XI ′, X1X

−1∞}

19

with I consisting of each of the 30 visible incidences involving those points and lines except one,(X−1

∞X1, A1), which must be derived.4 Note that, to write the intersection theorem itself, we canignore such objects as l which are incident with fewer than 3 other objects, since their existence isguaranteed by the projective plane axioms. (This in turn allows us to eliminate A∞ once l is gone.)In effect, we can “confine” the configuration. Our only card left is Desargues’ Theorem.

Now, if the intersection theorem is true, then X−1∞, X1, and A1 are collinear; conversely, this

is precisely what we need to prove the theorem. It is therefore equivalent to claim that X1, which isthe intersection of XI ′ and l′, is incident with the line X−1

∞A1. Similarly, we may treat X1 as theintersection of XI ′ and X−1

∞A0, so that the desired incidence is (X1, l′). We still have 29 assumed

incidences, and it is this intersection theorem that we shall verify as a consequence of Desargues’.5

Verification. Step 1. Relabel some of the configuration according to Figure 2.11 (a) (points X1

and X ′′ will not be important in this step). Triangles ABC and A′B′C ′ are perspective fromO, so by Desargues’ Theorem they are also perspective from some line l through each of P =(BC)(B′C ′), Q = (CA)(C ′A′), R = (AB)(A′B′).

(b)(a)

A

B

P

A'

B'

QR

C'

B'

A'

C

B

A

O

Figure 2.11: Left Multiplicative Inversion

Step 2. Relabel the original configuration according to Figure 2.11 (b), and relabel the pointsP , Q, and R as C ′, O, and C, respectively. Using the symmetry of Desargues’ Configuration(constructed in the last step) we know that triangles ABC and A′B′C ′ are perspective from O. ByDesargues’ Theorem again the points P , Q, and R are collinear.

The configurations corresponding to the remaining division ring axioms can alike be proven asconsequences of D [R, o]: commutativity of addition, associativity of both operations, and left andright distributivity. For instance, the configuration for (X ·Y ) ·Z = X ·(Y ·Z) is a bit more complex;

4Since A0−1 is undefined, it may seem necessary at first to include the stipulation that X 6= A0—that is, the

nonincidence relation X /∈ l′. And for all of these constructions we want to assume X /∈ l∞—i.e. X 6= A∞. However,the construction of either case will reveal a degenerate intersection theorem which is a consequence of the originalthree axioms. For now, we will ignore this technicality, though we will not be able to for long.

5Wrapping around the configuration in this fashion, we see that, as in Desargues’ Theorem, any of the 30 incidencesin the pictured configuration may be taken to be the conclusion, assuming the other 29.

20

Y·Z 1

X·Y!

Y 1Z 1

Y!

X!

(X·Y)·Z

X·(Y·Z)

X·Y

Y·Z

X Z Y

Figure 2.12: Multiplicative Associativity

we can only confine it to 13 points and 13 lines. The configurations for the distributive laws aremessier yet.

While it is satisfying to observe this structure emerge, the daunting configurations are notintuitively helpful. We impose a division ring structure on the “punctured” line l1, but surely wecan go further. Shouldn’t P be isomorphic to P2∆? In the next section we answer carefully in theaffirmative.

2.4 Homogeneous Coordinatization

As of now, trusting in some unwieldy configurational proofs, what we have is essentially an“x-axis” which we could assign the role of [0 : 1 : 0] under a new coordinatization scheme. Whatremains is to pick a “y-axis”, coordinatize the entire plane, and verify that linear equations definelines. While we could accomplish this using more horrific constructions, in the interests of spaceand understanding we will build a few more tools and apply them in a method adapted from [8].

Definition. A collineation is a configuration automorphism on a projective plane. The groupof collineations on a projective plane P is written Aut P. A collineation that fixes some line lpointwise and some point P linewise is called a projective collineation; l and P are then called theaxis and center of the collineation.

Examples (on D2). Let k be a nonzero element of D. The map σk : D2 → D2 defined by

σk(x :y :z) = (x :y :kz)

is a collineation taking each line [a :b :c] to [a :b :ck−1]. Furthermore, σk fixes every every point onthe horizon and line through the origin. It is therefore projective. When the axis and center of aprojective collineation are nonincident, as is the case with σk, the collineation is a homology.

Another projective collineation of D2 is the map τk, defined for any element k by

τk(x :y :1) = (x+ k :y :1)τk(x :y :0) = (x :y :0).

21

τk is the identity when k = 0, but otherwise τk takes each line [a :b :c] to [a :b :c− ak]. The axis ofτk is again [0 :0 :1], but its center (1 :0 :0) is incident with this axis. This makes τk an elation.

The “rotational” map ρ defined by

ρ(x :y :z) = (y :−x :z)

is another collineation, though since the only point it fixes is the origin (and the only line thehorizon) it is not projective.

The specific homologies and elations above will motivate the following choices. One conditionmust be met first.

Definition. Let PC[P,l](P) (or just PC[P,l]) be the subgroup of AutP of projective collineationswith center P and axis l. If m is any line through P , then PC[P,l] acts on [m], fixing P and theintersection point ml. If this action is transitive on the remaining points of m for every line m 3 P ,then P is said to be (P, l)-transitive.

l

m

P

A

A'

Figure 2.13: (P, l)-Transitivity

For instance, D2 is ((1 :0 :0), [0 :0 :1])-transitive since for any k ∈ D and line [0 :−y−1 :1] through(1:0 :0), τk takes (0 :y :1) to (k :y :1).

Lemma 2.1. A projective plane is desarguian ⇔ it is (P, l)-transitive for every point-line pair(P, l).

Proof. Look back on Desargues’ Configuration D (page 7). Take O to be our arbitrary point, o (theline through P , Q, and R) our arbitrary line, and p = OA our separate line through O. For thedirection (⇒) we want a (O, o)-collineation taking A to A′; for (⇐) we assume one exists. Eitherway, using the incidence axiom we may build up D with o = QR and P = (BC)(B′C ′), withoutthe incidence (P, o). Now if a map that takes A to A′ is to be a collineation, it need only send theother points B and C to B′ and C ′, since the points R and Q are fixed. (“Only”, because B andC can be any points off AA′.) Finally, since o is pointwise fixed, BC and B′C ′ must both containit. But this is exactly what it means for Desargues’ Theorem to hold!

This proof also provides uniqueness for the projective collineation sending A to A′, since theimages of all points off p are determined as drawn and the rest of p can be determined in likefashion from any other line, say q = OB. This existence-and-uniqueness is at the core of thefollowing discussion.

We now realize our binary operations from the previous section as collineations: Having desig-nated the quadrangle of reference and ∆ = l1 r {A∞} as before, for every P ∈ ∆ ( 6= A0 in thesecond case) let

τP ∈ PC[A∞,l∞], τP (A0) = P (“addition by P”)

22

andσP ∈ PC[A0,l∞], σP (A1) = P (“multiplication by P”).

Let us denote by PC[l1,l∞] the group of projective collineations with axis l∞ and center somewhereon l1. By the way, observe that, for Q′, Q, P,A∞ distinct on l1,

ς ∈ PC[P,l∞], ς(Q) = Q′ ⇒ ς = τP ◦ σR ◦ τP−1,

where σR is the homology sending τP−1(Q) to τP−1(Q′). That is to say, any element of PC[l1,l∞] isa PC[A∞,l∞]-conjugate of someone in PC[A0,l∞]. (In terms of addition and multiplication, what wemean is that exactly one linear transformation fixes p and takes q to q′, and it can be expressedy = (x− p)r + p, for some r.) It is clear that PC[A∞,l∞] ∩ PC[A0,l∞] = {id}, and the argument

σQ−1τPσQ(R) = R⇒ τP (σQ(R)) = σQ(R)

⇒ σQ(R) = A∞⇒ R = A∞

affirms that any conjugate σQ−1τPσQ is in PC[A∞,l∞], and hence that PC[A∞,l∞] is a normal sub-group of PC[l1,l∞]. We conclude that PC[l1,l∞] is a semidirect product of these two subgroups underconjugation,

PC[l1,l∞] = PC[A∞,l∞] o PC[A0,l∞].

We now use this group of collineations to rigorously define the operations −P = τP−1(A0),

P +Q = τQ(P ), P−1 = σP−1(A1), and P ·Q = σQ(P ) on ∆. (Again we must define multiplication

on the right, in order to agree with the general division ring case above—all because we chosehomogeneous coordinates via right multiplication way back on page 13.) Take a fresh look at thedivision ring axioms. ∆ inherits the ring operations + and · from PC[A∞,l∞] and PC[A0,l∞]. By theuniqueness in Lemma 2.1 we have

σQτPσQ−1 = τP ·Q,

since both send A0 to P ·Q. Right distributivity then follows from the equalities

(P +Q) ·R = τ(P+Q)·R(A0)= σRτP+QσR

−1(A0)= σRτQτPσR

−1(A0)= σRτQσR

−1σRτPσR−1(A0)

= τQ·RτP ·R(A0) = P ·R+Q ·R.

Even with our specialized tools, the remainder of the key result below requires a lengthy proof,the individual results of which will not be of use hereafter. As such, the full completion of this proof,including verification of the remaining division ring axioms and coordinatization of the plane, ispresented in Appendix A. In conclusion,

Equivalence 2.2. A projective plane P is desarguian ⇔ there is a division ring D s.t. P 'P2(D).

This equivalence will be the first in a string (or Jacob’s Ladder) of correspondences which willculminate to an expansive dictionary in Chapter 4. In the meantime, from this result we also obtaina “durer converse”:

23

Corollary 2.3. Any desarguian projective plane embeds (by collineation) into a projective 3-space.

Proof. Let D be the underlying division ring of P. Then P is isomorphic to the plane [0 : 0 : 0 : 1]in P3D.

Recall the aside from page 16. Observe in the picture that, it appears, the third diagonal point(AI)(X0X∞) does not lie on l1, in accordance with Fano’s Theorem. If it were so, however, thenthe points A0 and 2X would be one and the same. Remember that X was arbitrary on l1; we nowhave the language to interpret the old intersection theorem:

Lemma 2.4. The projective plane P2D satisfies Fano’s Theorem ⇔ D has characteristic 6= 2.

From now on we will restrict our discussion to desarguian projective planes and assume thateach has an underlying division ring.

24

3 More About Division Rings

A deeper discussion of these division rings is in order. In this chapter we shall characterize thecollineation group Aut P2D in terms of D3, introduce polynomials and polynomial identities, andthread these concepts together with intersection theorems in desarguian projective planes.

Example (a skew field). The pair of matrices

1 =(

1 00 1

), I =

(0 1

−1 0

)have some interesting features: They are linearly independent, in that a1 + bI = 0 for a, b ∈ Rimplies a, b = 0; and I2 = −1. Suspicious. The product of two elements in R1⊕ RI has the form(

a b−b a

) (c d

−d c

)=

(ac− bd ad+ bc

−ad− bc ac− bd

),

and the behavior of the complex numbers manifests. A bit more insight will reveal that R1⊕RI ' Cas division rings, with 1 and I playing the roles of 1 and i, respectively. Thus we have a representationof the complex numbers as 2× 2 matrices over the reals.

Continuing in this vein, consider the multiplicative group of the four matrices

1 =(

1 00 1

), I =

(0 1

−1 0

), J =

(i 00 −i

), K =

(0 ii 0

)and their negatives, which is called the quaternion group. These matrices are also R-linearly inde-pendent and span a subring

R1⊕ RI⊕ RJ⊕ RK = {(

a b

−b a

)| a, b ∈ C}

of M2C. In likeness to the complexes, this ring usually written as the 4-dimensional vector space

H = R⊕ Ri⊕ Rj ⊕ Rk. (3.1)

Its members are called the quaternions, and the basis elements 1, i, j, k satisfy the cayley table 3.1given by the matrices above. H× < GL2C as multiplicative groups, so H is in fact a division ring.

Table 3.1: The Quaternion Group

· 1 i j k

1 1 i j ki i −1 k −jj j −k −1 ik k j −i −1

25

The quaternions were named by their discoverer the mathematician Hamilton, and they aredenoted by his initial. We have completed the first two cayley–dickson constructions over R, towhich there is no end; but any further and our spaces cease to exhibit a division ring structure. Thequaternions, expressible as 2 × 2 matrices over the commutative complex numbers, will provide aconcrete example for much of the discussion in this chapter.

Written as in Equation (3.1), it is plain that H has dimension 4 over its center R; we writedimR H = 4. In this language any field F has dimension 1 over its center (itself); on the other hand,larger-dimensional, even infinite-dimensional, division rings exist as well. Throughout this chapterD will represent any division ring, F will denote its center, and ∆ will be the division ring of pointsin l1 r {A∞} isomorphic to D.

3.1 The Projective General Linear Group

Vector spaces also provide a medium in which to completely describe the collineations AutP2D,which we skirted in Section 1.2. (The group of automorphisms of a structure is a natural curiosityon its own, and our case as in others it provides vital condensation for some important results.)Think back to how P2D is defined: an equivalence relation (right multiplication) is imposed uponvectors in the space V = D3, and the resulting equivalence classes serve as points. Equivalently, wetreat D3 = D⊕D⊕D as the free right D-module of rank 3 and take points to be submodules ~vDof rank 1. We may refer to this right module as (D3)D.

Note. We can then take lines to be rank-2 submodules ~vD + ~wD spanned by two points, withincidence by containment; this is an alternative, concise way to define the projective plane over D,and we leave as an exercise in module theory that the two definitions agree—in a way that unifiesthe dualities of modules and of projective planes.

Looking for automorphisms of P2D, we have at our disposal the group of automorphisms of(D3)D—the general linear group GL3D of invertible 3 × 3 matrices with entries taken from D.(Now it is not immediately apparent that “invertibility” makes sense for matrices over divisionrings, since “left” and “right” inverses exist for a matrix A when the columns of A are rightlinearly independent (respectively, the rows of A are left linearly independent), as one can deducewith some thought. It is an important result that, indeed, A has a left inverse precisely when ithas a right inverse, and these inverses agree.)

Each matrix in GL3D imposes a linear transformation on V : Treating V as a column space,GL3D acts by left matrix multiplication:

A~v =

a11 a12 a13

a21 a22 a23

a31 a32 a33

v1v2v3

=

a11v1 + a12v2 + a13v3a21v1 + a22v2 + a23v3a31v1 + a32v2 + a33v3

.

Matrix multiplication naturally preserves addition, and

A(~uλ) = (A~u)λ (3.2)

for any λ ∈ D×, as can be routinely checked; so this action does indeed define a linear transforma-tion. (It is crucial that V be treated as column space under left multiplication by GL3D; treating~v as a row vector, (~vλ)A 6= (~vA)λ for noncommutative λ.)

26

It is also easy to check that GL3D is three times transitive on V , up to right linear independence:for vectors ~u,~v, ~w ∈ V the linear transformation described by

A = (~u | ~v | ~w) =

u1 v1 w1

u2 v2 w2

u3 v3 w3

is invertible when the vectors span (D3)D and takes ~e1 = (1, 0, 0) to ~u, ~e2 to ~v, and ~e3 to ~w. If A′

is defined similarly for independent vectors ~u′, ~v′, ~w′ then the transformation described by A′A−1

takes ~u,~v, ~w 7→ ~u′, ~v′, ~w′.The point is this: Every linear transformation A on (D3)D induces a collineation [A] of P2D,

as we will now see. For P2D = (A,L, I), (3.2) assures us that

[A] : A → A by A 7→ AA

is always a well-defined point map. Since A is invertible, this map is a bijection of A. Dually topoints, lines are equivalence classes of vectors living in the left D-module D(D3). Treated as rowvectors, they are then subject to linear transformations given by right multiplication by matricesin GL2D; each A induces a bijective line map

[A] : L → L by l 7→ lA.

In accordance with this treatment, denote l · A = ax + by + cz for a line l = [a : b : c] and pointA = (x : y : z) in the projective plane (a matrix product). Then (A, l) ∈ I ⇔ l · A = 0. We nowobserve that

(lA−1) · (AA) = 0 ⇔ l ·A = 0,

with A any matrix of GL3D. Thus is a collineation induced.

Definition. Since, for all points A ∈ A,

[A][B](A) = [A](BA) = A(BA) = (AB)A = [AB](A)

and therefore[A−1][A](A) = [I](A) = A,

we see that [ · ] : GL3D → AutP2D is a group homomorphism. So the set of collineations ofP2D induced in the preceding manner by matrices in GL3D is a subgroup of Aut P2D, called theprojective general linear group and denoted PGL3D.

So just how expansive is PGL3D? For starters, we know that it is transitive on the triangles ofP2D since GL3D is triply-transitive up to right linear independence. So look at the collineationsin PGL3D which fix the familiar triangle of reference A0 = (0 : 0 : 1), A∞ = (1 : 0 : 0), andA′∞ = (0:1 :0). These are induced by matrices of the form

A =

λµ

ν

,where the nonzero λ, µ, ν ∈ D are arbitrary. Broadening our view to Qref , we see that this [A] sendsA∗ = (1:1 :1) to the point (λ :µ :ν). Thus any point in P2D off the lines l1 = [0:1 :0], l∞ = [0:0 :1],and l0 = [1 : 0 : 0] can be the image of A∗ under a projective general linear transformation whichfixes the triangle of reference. Consequently,

27

Lemma 3.1. PGL3D is transitive on quadrangles; that is, if ABCD and WXY Z are two quad-rangles in P2D then there is a collineation in PGL3D which sends ABCD to WXY Z.

Proof. We already have collineations [A] and [W] which send the triangle of reference A0A∞A′∞ to

the triangles ABC and WXY , respectively. Let [D] and [Z] fix this triangle and take A∗ to [A]Dand [W]Z, respectively. Then

[W][Z][D]−1[A]−1 : ABCD 7→WXY Z.

Obviously the homomorphism [ · ] is not injective; any central scalar matrix in GL3D becomesthe identity in PGL3D. On the other hand, the previous discussion reveals that the kernel of thismap must consist of scalar matrices λI, since only they fix Qref . To fix all points of D, then, thescalar λ must satisfy

(λx, λy, λ) = λ(x, y, 1) = (x, y, 1)λ′ = (xλ′, yλ′, λ′) ,

for some λ′, given any pair x, y in D. Of course this requires λ′ = λ, hence that λ must commutewith everyone in D. Thus only λ in F× will do, giving us the central scalar matrices in GL3D,namely Z(GL3D). For concision, let us denote F× = Z(GL3D). Our description of PGL3D iscompleted by the Main Homomorphism Theorem.

Lemma 3.2. As groups, PGL3D ' GL3D�F× .

There are, as seen above, nonidentity collineations in PGL3D (namely, left multiplication bynoncentral elements of D) which fix Qref . In terms of coordinates, however, left multiplication isequivalent to conjugation:

(λx :λy :λz) = (λxλ−1 :λyλ−1 :λzλ−1) = (xλ :yλ :zλ),

and conjugation by a nonzero element in D is an inner automorphism on D. More generally, though,any automorphism ϕ of D naturally extends to (D3)D and, since automorphisms preserve 1, fixesQref . ϕ therefore induces a collineation on P2D by

(x :y :z) 7→ (ϕ(x) :ϕ(y) :ϕ(z)) and [a :b :c] 7→ [ϕ(a) :ϕ(b) :ϕ(c)].

On the other hand, suppose Φ is a collineation of P2D which fixes Qref . It follows that Φpreserves projectivities, and hence preserves the operations on ∆ from Section 2.3:

ΦA0 = A0,ΦA1 = A1,

−(ΦX) = Φ(−X),(ΦX)−1 = Φ(X−1),

(ΦX) + (ΦY ) = Φ(X + Y ), and(ΦX) · (ΦY ) = Φ(X · Y ).

Φ thus induces an automorphism on ∆—that is, an automorphism on D. Together with Lemma 3.1these facts complete our description of AutP2D.

28

Proposition 3.3. Let Stab(Qref) be the subgroup of AutP2D which fixes the quadrangle ofreference. Then

Stab(Qref) ' AutD.

Thus every collineation of P2D can be given exactly by some quadrangle (the image of Qref)and some automorphism of D.

Examples. C, a field, has no inner automorphisms other than the identity, so

AutP2C ' Aut C× PGL3C as groups.

Every automorphism of H is inner (see p. 19 of [11] for a proof), so

AutP2H = PGL3H.

Let us now apply this section to intersection theorems, the foci of our interest.

Theorem 3.4. Let C [A, l] be a (not necessarily universal) intersection theorem in a desarguianprojective plane P = (A,L, I), and suppose Q ⊆ C is a quadrangle. Let Q′ = PQRS ⊂ P be aquadrangle and have ΨQ denote the set of configuration isomorphisms of C into P which take Qto Q′.

Q ⊆ C↓ ↓ ψ

Q′ ⊆ C ′ ⊂ P

Assume that, for every such isomorphism ψ ∈ ΨQ, (ψ(A), ψ(l)) ∈ I. Then C [A, l] holds universallyin P.

Proof. Let τ : C → P be any configuration isomorphism. Say τ sends Q to Q′′ = P ′Q′R′S′ (aquadrangle), A to A′, and l to l′. We then want A′ ∈ l′.

By Lemma 3.1 there is a collineation ϕ : P → P which takes P ′Q′R′S′ to PQRS.

Q ⊆ Cτ ↓ ↓Q′′ ⊆ C ′′

ϕ l lQ′ ⊆ C ′

⊂⊂

P

This ϕ restricts to a configuration isomorphism ϕ |C ′′ from C ′′ to ϕ(C ′′) = C ′; say ϕ sends A′ toA′′ and l′ to l′′. Then

ψ = ϕ |C ′′ ◦τ : C → C ′

is a configuration isomorphism that sends Q to Q′, so ψ ∈ ΨQ. By our assumption, we must haveA′′ ∈ l′′. Since ϕ collineates, we also have ϕ−1(A′′) ∈ ϕ−1(l), or A′ ∈ l′.

This result simplifies drastically our criteria for universality by allowing us to restrict our field ofvision to the quadrangle of reference. It then also illustrates a close connection between (universal)intersection theorems of P2D and operations on D (viewed as projectivities). Finally, we reservethis very friendly consequence for later:

29

Corollary 3.5. Let C [A, l] be a (finite) intersection theorem, and let P and s be any point and linein the infinite desarguian projective plane P = (A,L, I). If (ψ(A), ψ(l)) ∈ I for every configurationisomorphism ψ : C → P under which ψ(B) /∈ s and Q /∈ ψ(m) for every point B and line m in C ,then C [A, l] holds universally in P.

Proof. Take any isomorphism ψ. If the points and lines of C ′ = ψ(C ) are all nonincident with Pand s then we are done; for the case that some are not, it will be sufficient to find a collineation of Ptaking every line away from P and every point off s. But this is equivalent to finding a collineationtaking P and s to some other point and line nonincident with the objects of C ′. By Lemma 3.1 weneed only show that such a point and line exist.

First suppose (P, s) /∈ I. Pick an arbitrary line n ∈ P. Find all intersections nm with m ∈ C ′—there are finitely many. So choose another point Q ∈ n; Q is then incident with no line of C ′.Dually, we can then find a line incident with no point of C ′ ∪ {Q}.

The case (P, s) ∈ I is left as an exercise.

Aside. The special unitary group SU2 consists of matrices P in GL2C for which detP = 1 (“spe-cial”) and P−1 = PH (“unitary”). It is easy to check that

SU2 = {(

a b

−b a

)| aa+ bb = 1},

which is isomorphic to the group of quaternions w + xi+ yj + zk with w2 + x2 + y2 + z2 = 1. Fora = w + xi and b = y + zi, the bijection(

a b

−b a

)↔ a+ bj = w + xi+ yj + zk

then interposes SU2 with the 3-sphere in R4; each point on the sphere gives a quaternion withmagnitude 1.

Conjugation by a quaternion reduces to conjugation by a member of SU2, and since R ∩ SU2 =Z(SU2) = {1,−1}, the action reduces further to conjugation by antipodal points of SU2. So denoteSU2�〈−1〉 by SU2

∗; this set “captures” the noncommutativity of H.∗ Given a point (p :q :r) in P2H,then,

SU2∗(p :q :r) = {(sp :sq :sr) | s ∈ SU2

∗} = {(sps−1 :sqs−1 :srs−1) | s ∈ SU2∗}

is the set of all images of (p :q :r) under Stab(Qref).Thus we might think intuitively of the projective plane P2H as the real projective plane P2R

together with a copy of the “unit 3-hemisphere” SU2∗ at every point. These would not be full

copies, however. For an affine point (x : y : 1), the sphere reduces to SU2∗�C(〈x,y〉), the set of left

multiplicative cosets of C(〈x, y〉), the centralizer of the subgroup generated by x and y. For infinitepoints (x : 1 : 0), the sphere reduces to SU2

∗�C(x) (over the centralizer of x), and for (1 :0 :0), as forall points (x :y :z) with x, y, z ∈ R—fixed by everyone in Stab(Qref)—the sphere reduces to 1. C

∗While SU2∗ is isomorphic to SO3 as seen in class, i thought it appropriate to remain in terms of antipodal points

of a sphere, a topic familiar to the paper.

30

SU2*

Figure 3.1: Intuitive Structure of P2H

3.2 Pappus’ Theorem

We can now strengthen Lemma 2.4 (which finally serves its purpose as a lemma). First let Fbe the configuration of a complete quadrilateral with diagonal points P,Q,R, so that F [P,QR] isthe intersection theorem that the diagonal points are always collinear.

Equivalence 3.6. P2D satisfies F [P,QR] (“P2D is nonfanoan”) ⇔ D has characteristic 2.

Proof. By Lemma 3.1, if the Fano Plane exists in P2D then it is ubiquitous in P2D; that is,F [P,QR] holds universally. Apply this to the inverse of Lemma 2.4.

Aside. Let F2[2X, l0] denote the intersection theorem above, using the notation from Figure 2.6.Since division rings can have any prime characteristic p, we can generalize Equivalence 3.6 tointersection theorems of the form Fp[pX, l0] using Theorem 3.4:

Equivalence 3.7. Let p be a prime. Then P2D satisfies Fp[pX, l0] (perhaps, “P2D is nonfanoanp”)⇔ D has characteristic p.

l'

4X 0

5X

2X!

2X 0

X!

2X4X

X 0

X

Figure 3.2: F5[5X, l0]

Theorem 3.4 says that the intersection theorem holds in P2D when it holds for any pointsA off Qref and X on l1 r {A0, A1, A∞}, since all other points and lines are uniquely determinedby Qref , A, and X. So, Fp[pX, l0] might also be phrased, “every harmonic sequence of points

31

has p members”. For example, Figure 3.2 depicts one construction 5X by taking 2X = X + X,4X = 2X + 2X, and 5X = 4X + X. (The lines of F5 are darkened.) There are several ways toconstruct this point, but we know from Equivalence 2.2 that the presence of Desargues’ Theoremmakes them all equivalent. C

Our prevailing goal is to catalogue and understand intersection theorems and their relationshipsto one another. For instance, the intersection theorems corresponding to the division ring axiomscan be (painstakingly) proven using Desargues’ Theorem. If an intersection theorem C [A, l] containsa quadrangle Q in C , Theorem 3.4 allows us to consider only isomorphisms taking Q to Qref . Asan example, the intersection theorem that X+Y = Y +X for arbitrary X,Y ∈ ∆ (Figure 3.3) canbe verified by applying D [R, o] twice, similarly to the verification that X−1 ·X = A1.

X+Y

YX

Figure 3.3: Additive Commutativity

There are worlds of intersection theorems that do not follow from Desargues’. The most renownedof these must be the original theorem of Pappus, a special case of Pascal’s Theorem for conics (whichwe do not tackle here). First we distinguish a hexagon, which consists of six ordered points P1, . . . , P6

(“vertices”) and the six lines (“edges”) of the form PiPi+1 (different from a “hexangle”).

Q PR

Z

X

CA

Y

B

Figure 3.4: Pappus’ Configuration

If the vertices of a hexagon lie alternately on two lines (both triples of mutually nonadjacentvertices are collinear), Pappus’ Theorem states that the (three) intersections of opposite sides lieon a line. In the figure, the hexagon AY CXBZ has opposite-edge intersections P,Q,R. We will seein a moment that this intersection theorem does not follow from Desargues’, though it does hold inthe euclidean plane whence it was derived. It can be shown, however (Hartshorne presents a 3-step

32

proof in [8]), that Desargues’ Theorem follows from Pappus’; every pappian projective plane is alsodesarguian.

This theorem becomes all the more important after the following observation: Consider theintersection theorem of Figure 3.5, which says that X · Y = Y ·X for any X,Y ∈ ∆. It is a specificcase of Pappus’ Theorem, though possibly in degenerate form (as when X = A0 is incident withAX1), but these cases are trivial in the plane. Thus the presence of Pappus’ Theorem in P2D,applied to each hexagon AX∞Y1I

′X1Y∞, implies that the multiplication in D is commutative.

I'

A

Y·X

Y 1

X!

X·Y

X 1

X

Y!

Y

Figure 3.5: Multiplicative Commutativity

Conversely, suppose we have a nondegenerate Pappus Configuration in P2D, with P,Q,Rnot necessarily collinear. From the configuration define the points E0 = (XZ)(PR) and E∞ =(AC)(PR). Lemma 3.1 then gives us a collineation ϕ of P2D sending E0E∞BY to A0A∞A

′∞A∗.

This collineation sends P and R to two points U and V on l1 = ϕ(PR), and ϕ(AC) = l∞. UsingA = A∗ for the constructions in Section 2.3, we also have ϕ(C) = U∞, ϕ(A) = V∞, ϕ(XZ) = l′,ϕ(Z) = U1, and ϕ(X) = V1. If D is commutative, then we know for certain that U∞V1, V∞U1, andl1 are concurrent, so by the inverse collineation ϕ−1 we get Q ∈ PR.

Equivalence 3.8. P2D is pappian ⇔ D is commutative.

We will denote Pappus’ theorem S1[S, l1], where S is the intersection of X∞Y1 and Y∞X1 inFigure 3.5. This notation will become clearer next section. For now, remember that S1 impliesD [R, o], while of course the converse is false. In the language of intersection theorems, we canrephrase Dehn’s original question as our main goal: Classify all C [A, l] such that

D [R, o] ; C [A, l] ; S1;

that is, classify the intersection theorems which lie “between” Desargues’ and Pappus’. Algebraistsgot extensive milage, and nearly achieved this goal, using algebraic relations like xy − yx = 0 of ageneralized type.

3.3 Polynomial Identities

To summarize the previous discussion: For the relationship xy − yx = 0 to be true for anysubstitutions x = ξ and y = η in D is for D to be a field. Let’s generalize this concept.

33

Let x = {x1, x2, x3, . . .} be an infinite collection of indeterminates (of which we will not needmore than countably many) and build the free Z-algebra over the alphabet x, written Z〈x〉, whichis the free Z-module over the set

{xi1 · · ·xin | ik ∈ N, n ≥ 0}

of words in the xi. The elements of Z〈x〉 are (noncommutative) polynomials; accordingly, Z〈x〉 ismade into a ring where words multiply by juxtaposition,

(αxi1 · · ·xim)(βxj1 · · ·xjn) = αβxi1 · · ·ximxj1 · · ·xjn .

Given a particular set of values ξ = {ξ1, . . . , ξn} in D and a ring homomorphism θ : Z → D,[some universal property of alphabet soup] declares a unique extension θ of θ to Z〈x1, . . . , xt〉, asubmodule of Z〈x〉, such that θ(xi) = ξi for each i ∈ [t]. Since any division ring D has an additiveidentity 0 and a multiplicative identity 1, the sensible homomorphism θ : Z → D is determined byθ(1) = 1; then

θ(±n) = ±n · 1 = ±(1 + · · ·+ 1) for n ∈ N.

When θ takes the xi to the ξi we call

θ : Z〈x1, . . . , xt〉 → D

the evaluation map at ξ. The image θ(f) of any polynomial is then the evaluation of f at ξ, andmay be written f(ξ); each f ∈ Z〈x〉 thus becomes a function from Dt to D for some t.

Definition. If f ∈ Z〈x1, . . . , xt〉 and f evaluates to 0 at every tuple in Dt, then f is said to be apolynomial identity of D. Adopting some familiar language, the identity f (or f = 0) is said tohold in D, and D is said to satisfy f .

Note. We depart here from the more standard discussion of polynomial identities, which considerspolynomials with coefficients in some field F which hold (or don’t) in division rings containing Fin their centers. Typically F is taken to be a prime field Π—either Q or some Fp, depending onthe characteristic of the division rings under scrutiny (each of which will contain Π ). Our reasonfor restricting to the integral identities from Z〈x〉 is that they make sense in a projective plane,as we will see. Our advantage is the ability to evaluate these polynomials in any division ring; infact, the strange-looking identities n = 0 (for n ∈ N) provide a litmus test for charD | n.

Example. s = xy− yx ∈ Z〈x, y〉, and D is a field when D satisfies s. This is one of the ubiquitousstandard identities, the nth of which is

Sn =∑σ∈Sn

(sgnσ)xσ(1) · · ·xσ(n)

in n indeterminates. (The homomorphism sgn : Sn → {1,−1} gives 1 when σ is even, −1 when σis odd.)

Suppose D satisfies S3; that is, for all ξ, η, ζ ∈ D,

ξηζ − ξζη + ζξη − ζηξ + ηζξ − ηξζ = 0.

34

We have 1 ∈ D as well, so for any ξ, η ∈ D (using ζ = 1) it is also the case that

0 = ξη − ξη + ξη − ηξ + ηξ − ηξ = ξη − ηξ;

so D satisfies S2. We leave as an exercise the general verification that D satisfies S2n whenever Dsatisfies S2n+1; we say that S2n+1 implies S2n. Clearly Sn implies Sn+1 for any n ∈ N; thence we callS2n and S2n+1 equivalent identities. It then makes sense to assign a division ring D p.i. (polynomialidentity) degree n when D satisfies S2n but not S2n−1. We will consider only the standard identitiesS2n from now on.

Identities As Intersection Theorems

Let’s take a closer look at Pappus’ Theorem S1. While S1 does hold in P2D exactly whenD is commutative, this is not exactly what the intersection theorem says. For now, let S1 referto Pappus’ Configuration without the incidence (S, l1)—the hypothesis of S1[S, l1]. To determinewhether S1[S, l1] holds in P2D, we only consider isomorphisms of S1 into P2D. Therefore, wedon’t consider the cases x = 0, y = 0, x = 1, y = 1, or x = y (which would cause the configurationS1 to degenerate in the plane). This is silly, since the identity S2 makes no such distinction. Wecan resolve this inconsistency by laying down precisely what we mean by the word “construct”(which we have used loosely up to this point) and by this strictness making up for a loosening ofour morphism requirement on intersection theorems.

First, we want to expand our definition of “configuration” to include point–line pairs A, l that areincident, pairs that are nonincident, and pairs that may be incident. So, in addition to incidences(A, l), let I also include nonincidences ���(A, l), meaning “A /∈ l”. The following definition is adapted,with subtle changes, from [10].

Definition. Define constructible configurations inductively by the following increments, tak-ing ∅ = (∅,∅,∅) to be constructible and remembering that no axioms must be met:

(i) If (A,L, I) is constructible and P /∈ A then (A ∪ {P},L, I) is constructible. (P is a newpoint.)

(ii) If (A,L, I) is constructible and m /∈ L then (A,L∪{m}, I) is constructible. (m is a new line.)

(iii) If (A,L, I) is constructible, m,n ∈ L, and m and n are incident with no common point, then(A ∪ {R},L, I ∪ {(R,m), (R,n)}) is constructible. (R is the intersection mn.)

(iv) If (A,L, I) is constructible, P,Q ∈ A, and P and Q are incident with no common line, then(A,L ∪ {o}, I ∪ {(P, o), (Q, o)}) is constructible. (o is the join PQ.)

(v) If (A,L, I) is constructible, P ∈ A, m ∈ L, and (P,m) /∈ I, then (A,L, I ∪ {����(P,m)}) isconstructible. (P is nonincident with m.)

Examples. As a configuration, Qref is constructible: Construct points A0 and A∞, then l1 is theirjoin. Construct the new point A′

∞ and stipulate that A′∞ /∈ l1. Then l∞ = A∞A

′∞ and l0 = A0A

′∞,

and the extra condition A0 /∈ l∞ gives us a nondegenerate triangle. Construct A∗ nonincident withthe three lines to get four triplewise noncollinear points. Construct m = A′

∞A∗, then A1 = l1m,and we are done.

35

Next, construct new lines mi and stipulate that A∞ /∈ mi. Then Xi = l1mi is a point on∆ = l1 r {A∞} with no other restrictions. Given several such points Xi on ∆, a point A off l1 andl∞, and a line l away from A, the perspective points Yi

A

[Xi on l are constructible.By the above examples, the operations on ∆ of Section 2.2 are all constructible.

Keep in mind that constructible configurations, like all configurations, are merely incidencestructures, and may not exist nondegenerately in P2D. Call an intersection theorem C [A, l] con-structible if C is constructible—or, more loosely, if 〈C [A, l]〉 = 〈C ′[A, l]〉 for some constructibleconfiguration C ′. (This allows for points X and Y on ∆ without the lines required to constructthem, and is necessary for constructing the hypothesis D of Desargues’ Theorem: Since each in-cremental construction appends either zero or two incidences to I, while D involves 29 incidences(without the claim (R, o)), it is impossible to construct D without using additional objects.) Theseuniversal constructible intersection theorems will be called u.c. intersection theorems.

We may now compromise somewhat the way we allow configurations to morph into planes.

Definition. Let C be a constructible configuration and ψ : C → C ′ ⊂ P2D a morphism. Callψ an evaluation (morphism) if ψ preserves incidence (as before) and nonincidence (i.e. ���(A, l) ∈I ⇒ ψ(A) /∈ ψ(l) in P2D). Thus, a morphism “degenerates” (is not an evaluation) if the imagesof nonincident objects of C are incident in the plane.

If C [A, l] is a u.c. intersection theorem, let us now say that C [A, l] holds in P2D provided forevery evaluation ψ : C → P2D, (ψ(A), ψ(l)) is an incidence of P2D.

In view of increment (v), we revise our concept of “confinement” to involve the removal oflines and points with at most two relationships (incidences or nonincidences). Observe that notall constructible configurations can be confined to ∅ (for instance, Qref). In the context of infiniteprojective planes, Corollary 3.5 keeps hold of the property that confinements do not affect thesatisfaction of an intersection theorem.

Expanding upon the examples above, each S2n now corresponds to a u.c. intersection theorem:From points X1, . . . , X2n on ∆ r {A0, A∞},1 construct the two sums∑

σ∈A2n

(Xσ(1) · · ·Xσ(2n)) and∑

σ∈S2nrA2n

(Xσ(1) · · ·Xσ(2n))

using the projectivities of Section 2.2. As in S1, let the lines slicing l1 at these two points intersectat a point S, which is not incident with l1 in the configuration. Name the configuration Sn; theintersection theorem Sn[S, l1] (hereafter abbreviated Sn) will then unite S and l1 in P2D preciselywhen the odd and even halves of S2n evaluate equally for any 2n-tuple from D. By Theorem 3.4we have

Equivalence 3.9. P2D satisfies Sn ⇔ D satisfies S2n.

(The Sn can be constructed in a variety of ways, producing as many distinct configurations,depending on the order of operations; by Equivalence 2.2 the presence of Desargues’ Theorem willmake all these variants equivalent.)

It follows thatS1 ⇒ S2 ⇒ · · · ⇒ Sk ⇒ Sk+1 ⇒ · · · .

1Since it will seldom again be necessary to refer to X0, X1, and X∞ from the operations on ∆, we can adopt theXi to stand for enumerated points in ∆.

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Note. It will simplify some future discussion to consider the additional identities S0, which, in theabove spirit, will be equivalent to S1(x1) = x1—which holds only in the trivial division ring {0}—and S∞ = 0, which holds in any division ring. Corresponding to these identities will be the inter-section theorems S0 = ∅[A, l], which holds only in the “trivial plane” P2{0} = ({A}, {l}, {(A, l)}),and S∞ = Qref [A0, l1], which holds in every P2D.

In the same manner as for the S2n, any polynomial f ∈ Z〈x1, . . . , xt〉 can be constructed onQref as a configuration, with, say, Pf the point on l1 which is the “evaluation” of f at the pointsX1, . . . , Xt by the operations on ∆. Let If denote the entire configuration constructed on Qref

by these operations. Since l0 is always incident with A0 in the quadrangle, we may then build anappropriate intersection theorem If [Pf , l0] with the following property:

Equivalence 3.10. D satisfies the polynomial identity f ⇔ P2D satisfies the u.c. intersectiontheorem If [Pf , l0].

3.4 Finite-Dimensional Division Rings

The standard identities Sn are one of the nicer generalizations of the commutative law xy = yxto division rings. For one thing, each intederminate xi appears exactly once in each term of Sn;this makes the Sn homogeneous, in that

Sn(αξ1, . . . , αξn) = αnSn(ξ1, . . . , ξn)

for any α ∈ F and ξi ∈ D. The Sn are also multilinear, in that

Sn(ξ1, . . . , λξi, . . . , ξn) + Sn(ξ1, . . . , λ′ξ′i, . . . , ξn) = Sn(ξ1, . . . , λξi + λ′ξ′i, . . . , ξn)

for any λ, λ′ ∈ D and i ∈ [n]; and they alternate [are antisymmetric?], in that

Sn(ξσ(1), . . . , ξσ(n)) = (sgnσ)Sn(ξ1, . . . , ξn)

(basically, any transposition of the indeterminates negates the polynomial).Let l be a linear expression α1ξ1 + · · ·+ αnξn with αi ∈ F and ξi ∈ D. As a consequence of the

aforementioned properties, observe that

Sn+1(ξ1, . . . , ξn, l) = Sn+1(ξ1, . . . , ξn,∑n

i=1 αiξi)=

∑ni=1 αiSn+1(ξ1, . . . , ξn, ξi) = 0

sinceSn+1(ξ, ξi) = (i n+ 1) · Sn+1(ξ, ξi) = −Sn+1(ξ, ξi)

forces each term to be zero. Thus the symmetric identities detect linear dependence, a usefulproperty when working in vector spaces.

Lemma 3.11. Every division ring of finite dimension over its center satisfies a standard identity.

Proof. Say dimF D = n and consider the identity Sn+1 evaluated at any ξ1, . . . , ξn+1 ∈ D. Asvectors, the set of ξi must then be linearly dependent, so we may assume (via some change insubscripts) that ξn+1 = a1ξ1 + · · ·+ anξn = l(ξ1, . . . , ξn). It follows from the above discussion that

Sn+1(ξ1, . . . , ξn, l) = 0.

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Thus the quaternions satisfy S5, hence also S4 (since 5 is odd). Since they are noncommutative,however, they do not satisfy S2, hence nor S3. So p. i.deg H = 4

2 = 2.A major aim of polynomial identity theory is to characterize the set of polynomial identities

satisfied by Mn(F ), the n × n matrices over a field F (or, more generally, a commutative ring).Any identity of Mn(F ) is also satisfied by the division rings with representations in Mn(F ), so thisendeavor has gone far toward the classification of meso-desarguian–pappian intersection theorems.

Example (Wagner). Recall the Cayley–Hamilton Theorem, that any n× n matrix M satisfies itscharacteristic polynomial. In the 2× 2 case the theorem states that, for any matrix M ,

M2 − (trM)M + (detM) = 0;

note that when trM = 0 it gives M2 = detM , a scalar matrix, which commutes; hence [M2, X]vanishes when M is without a trace. Using the commutativity of F , for any two matrices Y and Zthe matrix Y ·Z−Z ·Y has a diagonal of 0s; so tr[Y, Z] = 0. Taken together, we obtain the identity

[x, [y, z]2] = 0, or x · [y, z] = [y, z] · x,

which holds in any 2× 2 matrix ring over a field, including H.

Wagner expanded upon this idea to produce the collection of polynomials

Wm(x, y) = Sm[y′′, y′′′′, . . . , y(2m)],

where a′ = [x, a], with the property that Wm holds in Mn(F ) whenever 2m > n(n− 1).The inverse of Lemma 3.11 is due to Kaplansky:

Theorem 3.12 (Kaplansky). A division ring of infinite dimension over its center satisfies nopolynomial identities over its center.

Among the gems of noncommutative algebra is the fact that the dimension dimF D of a divisionring over its center is necessarily either infinite or a perfect square. A pleasant result published in1950 provides a second litmus test:

Theorem 3.13 (Amitsur–Levitzki). S2n holds in D ⇔ dimF D ≤ n2.

This means that p. i.degD is always the square root of dimF D, and to compare dimensions ofdivision rings is to compare p.i. degrees. By now the proofs of many results are outside the scope ofthis paper, and will be omitted. For another example, note that Theorems 3.12 and 3.13 allow us toreduce our field (or division ring) of vision to finite p.i. degree—that is, finite-dimensional divisionrings, and most importantly those that are noncommutative. This draws out another result:

Theorem 3.14 (Wedderburn). Every finite division ring is a field.

Thus if we want 1 < dimF D < ∞ then we need D infinite, hence F infinite as well. That is,every finite-dimensional noncommutative division ring either contains a copy of Q or features aninfinite-degree field extension over Fp, depending on the characteristic of D.

Remember that, for m an integer, the polynomial m ∈ Z〈x〉 is an identity of D precisely whencharD | m. In [1] Amitsur combined such identities with the standard identities S2n (0 ≤ n ≤ ∞)

38

to create the following descriptive polynomial identities: For 0 ≤ t ∈ Z, let p = (p1, . . . , pt) be atuple of distinct primes and n = (n0, . . . , nt) a nondecreasing tuple of natural numbers, 0, or ∞,where 0 ≤ n0 ≤ · · · ≤ nt ≤ ∞. Then define

S(p;n) = m0S2n0(x0) +m1S2n1(x1) + · · ·+mtS2nt(xt),

where m0 = p1p2 · · · pt, mi = m0pi

for i ≥ 1, and xi = {xi,1, . . . , xi,2ni}—a polynomial identity in∑ti=0 ni indeterminates. If t = 0 then p = ∅ and S(∅; {n0}) = S2n0(x0). Amitsur proved, as is not

difficult to infer,

Theorem 3.15. D satisfies S(p;n) ⇔ either

(i) charD 6= pi for any i ∈ [t], in which case p. i.degD ≤ n0; or

(ii) charD = pi for some i ∈ [t], in which case p. i.degD ≤ ni.

Amitsur went further, showing these identities to be a complete set of equivalence class repre-sentatives from Z〈x〉:

Equivalence 3.16. Every polynomial identity is equivalent to some S(p;n).

For our purposes, it follows that for every intersection theorem If (with f ∈ Z〈x〉) there are p andn such that If holds in P2D ⇔ S(p;n) holds in D.

A concise criterion for two division rings to share polynomial identities is easy to see: Let D andD′ be division rings (of the same characteristic), then D satisfies all polynomial identities of D′ ⇔p. i.degD ≤ p. i.degD′ (≤ ∞). This is tied to a broader result from [1], that two division rings Dand D′ of the same p.i. degree (and characteristic) satisfy the same set of rational identities, andclearly that the converse is true.

By their nature rational expressions, which involve the operation −1 (last seen performed on ∆),are difficult to define, and especially to classify as “rational identities”. The usual setup goes likethis (examples to follow): Let y = {y1, . . . , yt} be a finite collection of additional indeterminatesand build the free algebra Z〈x;y〉. For each i ∈ [t] let pi(x; y1, . . . , yi−1) be an integral polynomial.Let I0 be the two-sided ideal in Z〈x;y〉 generated by the polynomials (yipi−1), i ∈ [t], and take thequotient ring Z〈x;y〉/I0. Effectively we have defined the yi to be inverses of the pi; the expressionpt becomes a rational identity of D when each admissible assignment ϕ : x → D (whose extensionto each Z〈x;y〉/(y1p1 − 1, . . . , yi−1pi−1 − 1) sends pi to a nonzero value in D) finally extends toϕ : Z〈x;y〉/I0 → D taking pt to 0. The nonnegative number t is a bound on the distortion numberof pt, which can be thought of as the greatest number of “nested” inverses.

We call two division rings rationally equivalent when they satisfy the same set of rational identi-ties, and Amitsur’s result nails down the equivalences classes as the sets of division rings of commoncharacteristic and p.i. degree. Consequently, we will discuss rational identities holding or not hold-ing “in Dp,n”, meaning “in division rings of characteristic p and p.i. degree n”. The task is then todetermine exactly what distinctions a given rational identity might make—what information abouta division ring can be extracted from the knowledge that it satisfies a particular identity.

For a time it seemed as though all the information about a division ring extractable by anyrational identity could be extracted just as well with an S(p;n). Happily, this is not so, and therealm of rational identities is rougher terrain: a gap unnoticed by polynomial identities becomesdiscernible by the rational identities, as first described in [3] and further illuminated in [4]. Thefinal chapter will bring our theory of identities up to speed and bind together our dictionaries.

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4 Rational Identities

We are working toward a fundamental unification of identities with intersection theorems; justas we required an iterative process to generate the set of constructible configurations, we nowintroduce an iterative process which will generate the collection of rational expressions. Just asdifferent configurations may yield a common point in any plane (e.g. X + Y and Y + X), thefollowing definition will yield distinct expressions which we know intuitively to be the same.

Definition. Let x = {x1, x2, x3, . . .} be an infinite collection of indeterminates and consider theset R[x] of rational expressions in the xi, defined inductively as follows:

• 0 and 1 are expressions;

• each xi is an expression;

• if e and f are expressions, then so are −e, e + f , e−1, and e · f (call these increments of eand f).

Call e a subexpression of f if f is an increment of e or, iteratively, if some subexpression of f is anincrement of e. Note that all expressions are distinct, e.g. xi + xj 6= xj + xi, and no expressions areexcluded, e.g. 0−1 ∈ R[x].1

For any elements ξ = ξ1, . . . , ξt of D, define the partial map

ϕξ : R[x1, . . . , xt] 99K D,

to be written ϕξ(e) = e(ξ), also inductively by

• 0(ξ) = 0 and 1(ξ) = 1;

• each xi(ξ) = ξi;

• if e(ξ) = α, f(ξ) = β then (−e)(ξ) = −α, (e+ f)(ξ) = α+ β, and (e · f)(ξ) = α · β;

• if e(ξ) = α 6= 0 then (e−1)(ξ) = α−1; if α = 0 then (e−1)(ξ) is undefined;

• if e(ξ) is undefined and f has e as a subexpression then f(ξ) is undefined.

Call this map the evaluation of R[x1, . . . , xt] (and each e(ξ) the evaluation of e) at ξ.Denote by e(D) the set of evaluations of e at tuples in D. Call e ∈ R[x] a rational identity of

D when e(D) ⊆ {0}. (e is a degenerate identity when e(D) = ∅.) For our purposes it will help toalso consider the statement e : eL = eR a “rational identity” when eL − eR is a rational identity inthe previous sense.

Examples. For every polynomial f ∈ Z〈x1, . . . , xt〉 there is at least one rational expression f ′ ∈R[x1, . . . , xt] (and usually many more) for which f ′(ξ) = f(ξ) for every ξ ∈ Dt. For instance, wecan redefine the standard identities as the rational identities

S2n :∑σ∈A2n

(xσ(1) · · · · · xσ(2n)) =∑

σ∈S2nrA2n

(xσ(1) · · · · · xσ(2n)),

1We do, however, reserve the privilege to write e − f for e + (−f), e2 for e · e, and [e, f ] for e · f − f · e withoutambiguity.

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where the sums are taken in order of some prechosen enumeration of S2n. (When not specified, anordered sum or product will be taken from left to right.)

For any rational expression g in Z〈x;y〉/I0, too, there is a rational expression g′ in R[x] forwhich g′(ξ) = g(ξ) for every ξ from D at which both are defined, and neither is defined without theother. When g′ is built in the natural way, the distortion number of g gives the maximum numberof iterations −1 in any chain of subexpressions of g′. So, if e is any expression in R[x], it makessense to call this number the “distortion number” δ(e).

Recall Wagner’s identity [x, [y, z]2] = 0 for 2× 2 matrices. As a special case, the expression

θ(x, y) = [x, [x, y]2] · [x, [x, y]−1]−1,

with distortion number δ(θ) = 2, is a (nondegenerate) rational identity for 2 × 2 matrices (andhence for H) but not for 3 × 3 matrices. In fact, when θ(ξ, η) is defined in M3(F ) it evaluates tothe scalar matrix det[ξ, η]. This fact proves to be quite useful.

Hua’s identityH : (x−1 + (y−1 − x)−1)−1 = x− x · y · x

is nontrivial, in that it does not reduce to 0 = 0 in the ring Z〈x, y; ·〉/I0. However, it is satisfied byevery division ring (left as an algebraic exercise).

4.1 Completing the Picture

As was not the case with Z〈x〉 and its ideals, speaking in terms of R[x] is quite conducive togeometric interpretation: By incrementing, it is possible to write, for any rational expression e, anincremental sequence of expressions

{x1, . . . , xt, e1, . . . , en = e}

in which each ei is an increment of one or two of the expressions before it. (In practice we willlook only at {e1, . . . , en}.) This sequence should remind us of the sequences of constructions inthe last part of Section 3.1; in fact, the increments correspond precisely with he operations on∆. While order of addition and multiplication matters to both, neither notices the order in whichsubexpressions are built; for instance, whether e is completed before or after f does not affect thestructure of e+ f , but e+ f is different from f + e.

Together with Lemma 3.1 this property gives us half of our fundamental result: Let e : eL = eRbe a rational identity in expressions ei, i = L,R. Construct the configuration Qref with a set ofpoints X = {X1, . . . , Xt} on ∆. Have Iei denote the configuration produced by the correspondingoperations on these points, together with Qref . (Note that both of, and only, these steps requireconstruction (v): eachXi must be off l∞, and each inverted subexpression f(X) must be off l0.) Haveei(X1, . . . , Xt) = ei(X) refer to the “evaluation point” on l1, which is sliced by the last constructedline lei . (See Figure 4.1.) Let Ie = IeL ∪IeR ; then assign E = leL leR , a point nonincident with l1in the configuration. Then the u.c. intersection theorem Ie[E, l1] makes the claim that eL(X) andeR(X) are the same point. (The points ei(x) are unnecessary to Ie, but referenced for clarity.)

The Integral Identity–U.C. Intersection Theorem Dictionary. D satisfies the rational iden-tity e⇔ P2D satisfies the u.c. intersection theorem Ie[E, l1]. Moreover, e and Ie[E, l1] degeneratetogether.

41

X 2 X 1 e i(X)

l e i

I

I'

A

Figure 4.1: The Configuration Ieiof Expression ei

All that remains to be verified is mutual nondegeneracy; what this means is that e cannot beevaluated in D precisely when Ie cannot be evaluated in P2D.

Proof of “Moreover”. D satisfies e degenerately when, for every ξ from D, some inverted subex-pression f of e evaluates to 0. On the other hand, P2D satisfies Ie[E, l1] degenerately when, foreach set of points X on ∆ ⊂ P2D, some nonincidence ����(P,m) fails, i.e. some subconfiguration ofI unites P and m. But the only nonincidences enforced are �����(Xi, l∞) (which can be avoided bythe infiniteness of ∆, and do not correspond to evaluations of e in D) and �����(f(X), l0) for somesubexpression f of e which is inverted in e. Thus e holds degenerately in precisely those D forwhich I [E, l1] holds degenerately in P2D.

Finally, we have a watered-down version of the main theorem from [3], which gives for rationalidentities the criterion corresponding to Theorem 3.4 for polynomial identities.

Theorem 4.1 (Amitsur; Bergman). Fix p (0 or prime) and let m,n ∈ N, then Dp,m satisfies everyrational identity of Dp,n if and only if m | n.

Finding “Exotic” Identities

So what do these identities look like? How messy do they get? How messy must they get? Untildeclared otherwise, hold p, our characteristic, fixed.

Example (Bergman). In 1974 was unveiled the first rational identity to fill a “divisibility gap”:Recall θ(x, y) from before, an identity of p.i. degree-2 division rings which evaluates to det[ξ, η]for substitutions ξ, η in M3(F ). In the paper Bergman used properties of the determinant and ofcommutators to great effect (see Section 6 of [3] for a full, accessible discussion) to produce theexpression

B(x, y) = θ(y) · θ(y′) · θ(y′′−1) · θ(y′′′−1)

(using the recent notation a′ = [x, a]), which evaluates to 1 in M3(F ) where defined, which doesoccur (and of course to 0 in M2(F )); thus the identity

B : B(x, y) = 1

42

holds in Dp,3 but not in Dp,2.Expanding the expression, B(x, y) has the form

(y′2)′

(y′−1)′· (y′′2)′

(y′′−1)′· ((y′′−1)′2)′

((y′′−1)′−1)′· ((y′′′−1)′2)′

((y′′′−1)′−1)′,

where each θ(a) = (a′2)′

(a′−1)′ evaluates to a commuting scalar matrix in M3(F ). We may apply this tothe expression to the four θ(a) of B obtain the equivalent form

B : (y′2)′ · (y′′2)′ · ((y′′−1)′2)′ · ((y′′′−1)′2)′ = (y′−1)′ · (y′′−1)′ · ((y′′−1)′−1)′ · ((y′′′−1)′−1)′,

where both sides are equal (and not always zero) for evaluations in Dp,3 while the left side is zeroand the right side not necessarily zero for evaluations in Dp,2.

And the search was on. . . .

Example (Rowen). In [10], Rowen filled the next gap with an identity holding in division rings ofp.i. degree 4 and not in those of p.i. degree 3. This one requires the fifth Capelli Polynomial

C5(x1, . . . , x5) =∑σ∈S3

(sgnσ)xσ(1)x4xσ(2)x5xσ(3)

and a more general likeness of θ defined by

f(x) = f(x1, x2, x3) = [[x1, x2]2, x3] · [[x1, x2], x3]−1.

This expression f has the property that, for any ξi ∈ D with p. i.degD = 4, either f(ξ)2 ∈ For f(ξ)4 ∈ F + Ff(ξ)2; thus {1, f(ξ)2, f(ξ)4} is an F -linearly dependent set under all evalua-tions. The Capelli Polynomials have a way of detecting such dependence (similarly to the standardpolynomials), which doesn’t occur in Dp,3, and Rowen presents the identity

R : C5(1, f(x)2, f(x)4, x4, x5) = 0,

which for the purposes of constructing intersection theorems would be better written∑σ∈A3

f(x)2σ(1)−2x4f(x)2σ(2)−2x5f(x)2σ(3)−2 =∑

σ∈S3rA3

f(x)2σ(1)−2x4f(x)2σ(2)−2x5f(x)2σ(3)−2.

In 1991 Le Bruyn [4] published what seems to be the first strict condition on a rational expressionthat vanishes in division rings of p.i. degree n but not in those of p.i. degree m - n, namely

Proposition 4.2 (Le Bruyn). Fix p and let m,n ∈ N with m - n. If e ∈ R[x] is an identity of Dp,n

but not of Dp,m then e has distortion number δ(e) ≥ b nmc.

For instance, δ(B) = 1 + δ(θ) = 3 > 1 = b32c while δ(R) = 1 = b4

3c. Le Bruyn achieved a muchstronger result, a version of which we will see without regard to distortion numbers.

Definition. For n = {n1, . . . , nl} a set of distinct natural numbers, call n ∈ N n-reachable ifthere is a young diagram in row lengths from n with n boxes; i.e. if

n =l∑

i=1

aini

has a solution for the ai in N ∪ {0}.

43

Example. 23 is not {5, 7, 15, 24}-reachable. For, if it were, it would obviously be {5, 7, 15}-reachable;and 23− 15 = 8 is not {5, 7}-reachable, so 23 would have to be {5, 7}-reachable. But the diagramsfor the possible numbers of blocks of size 7 all fail:

�������������������������

���������������������������

������������������������

��������������������������

.

n-unreachability is a looser condition than gcdn - n, since the ni may only be added.

Theorem 4.3 (Le Bruyn). Let n = {n1, . . . , nl} ⊂ N and n ∈ N. Then there is a rational identitywhich holds in Dp,n but not in any Dp,ni if and only if n is not n-reachable.

Such an expression is an identity of any division ring of characteristic p and dimension n2 overits center, but not of any division ring of characteristic p and dimension ni

2, for any i ∈ [l]. Notethe special cases with n = ∅ for 0 ≤ n ≤ ∞, exemplified by the familiar S2n.

P.I.-Degree Arithmetic

We can amalgamate Le Bruyn’s Identities in a fashion similar to Amitsur’s, possibly to com-pletely characterize the information accessible about a division ring D from a rational identitywhich it is known to satisfy. This in turn would determine the specific properties of D discerniblefrom any meso-desarguian–pappian intersection theorem created via Equivalence 4.1. (Such a clas-sification in the literature, as was done for polynomial identities in [1], has eluded the author.)First, let us tighten our description of Le Bruyn’s identities.

Definition. For any subset n ⊆ N, let the span of n be the set of all n-reachable numbers.Conversely, if n /∈ spann and the ni < n then call n n-deficient. There then exist maximal n-deficient sets for any n ∈ N. If e is a rational identity, its satisfaction sequence will be the binarysequence (b1, b2, b3, . . .), where

bj ={

0 if e holds in Dp,j

1 if not.

The satisfaction sequence for the product e · f is then the componentwise product of the sequencesfor e and f . (This uses the known fact that, given expressions e, f ∈ R[x] which are not identitiesfor a division ring D—that is, there are ξ,η ⊂ D such that f(ξ), g(η) 6= 0—there is ζ ⊂ D suchthat (f · g)(ζ) = f(ζ)g(ζ) 6= 0, and hence f · g is not an identity for D.)

Corollary 4.4. Given n ⊂ N and n ∈ N n-unreachable, there exist rational expressions Tn,n whichgive 0 in precisely those Dp,m for which m ∈ [n] r spann.

The point is to distinguish between such “Le Bruyn identities” as T{3},7 and T{3,5},7—to havethe former give 0 in Dk,5 and the latter not. That is, we wish to decree that these identities havethe satisfaction sequences

T{3},7 : (0, 0, 1, 0, 0, 1, 0, 1)T{3,5},7 : (0, 0, 1, 0, 1, 1, 0, 1)

instead of the ambiguousT{3},7 : (0, 0, 1, 0, ?, 1, 0, ?, ?, . . .)T{3,5},7 : (0, 0, 1, 0, 1, 1, 0, ?, ?, . . .).

44

We know of the 0s in both because {3, 6} can only be appended with 5 and stay 7-deficient ({3, 5, 6}is maximal). By appending {8, 10} to {3} or {3, 5} we don’t change 7-reachability at all but forcebm = 1 for any m > 7. We can then force T{3},7 to vanish on Dk,5 by multiplying by T{3},5, whichthen gives the satisfaction sequence

T{3},7 : (0, 0, 1, 0, 0, 1, 0, 1).

The proof will merely generalize these tactics.

Proof of the corollary. Let n be n-deficient, then by Le Bruyn’s theorem we have an expression“en,n”, which holds in Dp,n but not in the Dp,ni . Take n1 to be the smallest member of n. Append{n + 1, . . . , n + n1} to n to get n′, which is also n-deficient, then every natural number greaterthan n is n′-reachable. The new en′,n holds only in Dp,m where m | n and possibly for otherm ∈ [n] r spann.

For each m < n not dividing n and outside spann, we want Tn,n to hold in Dk,m as well. So letm = n∩ [m] and multiply en′,n by em,n. Since m is m-deficient, the satisfaction sequences for thesetwo identities agree below m. (By “agreement” we mean that both are 0, both are 1, or both areambiguous, unknown. For a stricter argument, start with the first ambiguity and use induction.)The latter sequence is uniformly 1 above m, so the product sequence agrees with that of en′,n exceptthat bm is definitely 0.

The product is what we’ll call Tn,n. It holds in precisely those Dp,m with m ∈ spann∩ [n], whichis what we wanted.2

The next task is to reduce every rational identity to such a form.

Proposition 4.5. Let e ∈ R[x], then there exist n ⊂ N and n ∈ N not n-reachable such that, as mranges over N, e holds in Dp,m precisely when Tn,n holds—equivalently, e and Tn,n have the samesatisfaction sequences.

Proof. If e is satisfied by Dp,n for no (respectively, all) n ∈ N then use T∅,0 = S0 (respectively,T∅,∞ = S∞). Else, let n1 be the first natural number for which Dp,n1 dissatisfies e and let δ be thedistortion number of e. By Proposition 4.2, m > δn1 implies Dp,m dissatisfies e, so there is a lastnatural number n for which Dp,n satisfies e.

The set of natural numbers that yield satisfaction is thus a subset of [n], and by the arithmeticabove its complement (to be called n) is n-deficient. Thus the Le Bruyn identity Tn,n has the samesatisfaction sequence as e.

Return to the general case, in which an expression e ∈ R[x] can be evaluated (or is undefined)in the Dp,n, as p ranges over 0 and the primes and n ranges over N and ∞. To each expression therecorresponds a “satisfaction array” of 0s and 1s, with columns indexed by N ∪ {∞} and rows by 0and the primes. Bootstrapping off Amitsur’s S(p;n) and Le Bruyn’s Tn,n, we present a collectionof “Amitsur–Le Bruyn Identities”

T (p; n,n) = m0Tn0,n0(x0) +m1Tn1,n1(x1) + · · ·+mtTnt,nt(xt),

2Le Bruyn’s original identities have the property that their distortion numbers are minimal with respect to thedistinctions they accomplish; we appear to lose this property under these tweaks, but remember that distortion numberis not too strongly correlated with the number of iterations required, a measure we are more eager to minimize.

45

where p = (p1, . . . , pt) is a tuple of primes (with the mi defined as on page 39), n = (n0, . . . , nt)is a nondecreasing tuple of natural numbers or 0 or ∞, n = (n0, . . . ,nt) is a tuple of subsets of N,each ni is not ni-reachable, and the Tni,ni are integral identities specific to fields of characteristicpi (with p0 = 0). Naturally arises

Question 4.6. Is every e ∈ R[x] equivalent to some T (p; n,n)?

If this is the case, then the satisfaction arrays for all (integral) identities—hence all (u.c.) intersectiontheorems—are completely classified.

4.2 Intersection Theorems As Identities

Suppose a u.c. intersection theorem C [A, l] holds nondegenerately in P2D—that is, it holdsfor every evaluation of C , and at least one exists. What does this tell us about D? Consider anevaluation of C in the plane, using homogeneous components, in terms of the increments fromSection 3.3 that build up C .

(i) The point P must have homogeneous coordinates (xP :yP :zP ), not all zero.

(ii) The line m must have homogeneous coefficients (am :bm :cm), not all zero.

(iii) The homogeneous coordinates (xR :yR :zR) of intersection R must also satisfy

am · xR + bm · yR + cm · zR = 0 = an · xR + bn · yR + cn · zR.

(iv) The homogeneous coefficients (ao :bo :co) of join o must also satisfy

ao · xP + bo · yP + co · zo = 0 = ao · xQ + bo · yQ + co · zQ.

(v) The homogeneous components of P and m must also satisfy

m · P = am · xP + bm · yP + cm · zP 6= 0.

If C is thus evaluated, the intersection theorem C [A, l] then states that, given these conditions, itmust follow that

l ·A = al · xA + bl · yA + cl · zA = 0.

Using this correspondence, we can read any u.c. intersection theorem as a rational identity, in amanner adapted from [1] and [10].

Construction 1. Let C [A, l] be a u.c. intersection theorem and set up a finite sequence

∅ = C0 ⊂ C1 ⊂ · · · ⊂ Ck−1 ⊂ Ck = C

of incremental configurations taking ∅ to C (that is, each Ci+1 is an increment of Ci). Let f∅be the expression 1, defined and nonzero for any evaluation in D. Eventually we will arrive atan expression fC , in terms of indeterminates corresponding to the coordinates and coefficientsof points and lines in P2D, which is defined and nonzero whenever the points and lines form a

46

(nondegenerate) evaluation of C in the plane. To make an identity, then, we will simply multiplyfC by l ·A.

At each increment, we take Ci to Ci+1, and assume fCiexists and is nonzero for any evaluation

of Ci in P2D and any nondegenerate assignments for the other indeterminates involved. Give eachpoint P coordinates (xP : yP : zP ) and each line m coefficients [am : bm : cm], where xP , . . . , cm arerational expressions, possibly indeterminates.

(i) LetfCi+1

= fCi· (vi+1 · P )−1,

where the components of vi+1 = [vi+1,1 :vi+1,2 :vi+1,3] are distinct and appear nowhere in fCi.

vi+1 · P = 0 when all the components of P are 0, so (vi+1 · P )−1 exists only if P is a validpoint—and for any valid point there are some assignments of the vi+1,j for which vi+1 ·P 6= 0.(Think of vi+1 as some line in P2D nonincident with P .) Thus fCi+1

exists and is nonzero forany evaluation of Ci+1 in P2D, for some assignments of vi+1.

(ii) LetfCi+1

= fCi· (m · Ui+1)−1,

where the components of Ui+1 = (ui+1,1 : ui+1,2 : ui+1,3) are distinct and appear nowhere infCi

.

Analogously, fCi+1exists and is nonzero for any evaluation of Ci+1 in P2D, for some assign-

ments of Ui+1.

(iii) LetfCi+1

= fCi· (vi+1 ·R)−1,

where the components of vi+1 are again new indeterminates but the components of R areinstead of the following forms:

xR is a new indeterminate;yR = (cm−1 · bm − cn

−1 · bn)−1 · (cn−1 · an − cm−1 · am) · xR;

zR = −cm−1 · (am · xR + bm · yR).

These assignments come from “solving” for the components of R in the equality m ·R = 0 =n · R, and indeed for any evaluation of the indeterminates in D for which these expressionsare defined, they must evaluate to zero, as can be seen algebraically. The inverse of (cm−1 ·bm − cn

−1 · bn) prevents us from trying to find the intersection of a line with itself. (It maystill give zero for two different lines, distinct at the a-coefficient, but we will tackle this quirkbelow.) The (vi+1 ·R)−1 then serves to eliminate the nonsense case R = (0:0 :0).

(iv) LetfCi+1

= fCi· (o · Ui+1)−1,

dually to the intersection case.

(v) LetfCi+1

= fCi· (m · P )−1.

Since the components of P and of m have already been “forced” not all zero, this factor servesonly to remove the case P ∈ m from consideration.

47

Finally, once fC is obtained, take

fC [A,l] = fC · (l ·A),

which will evaluate to zero whenever fC is defined (assuming, of course, that the intersectiontheorem holds in P2D).

One problem manifests from this method: Thanks to (iii) and (iv), no zP or cm is safe from the−1 operation; but this is easy to remedy. If the intersection theorem holds for all evaluations in Dfor which these third components are nonzero, then the remaining cases always have a point on theline l∞ = [0:0 :1] or a line through the point A0 = (0:0 :1); Corollary 3.5 gives us a collineation ϕof P2D reducing this to the case that no one is incident with l∞ or A0, thence ϕ(A) ∈ ϕ(l) in thenew evaluation, hence A ∈ l in the first place.

The U.C. Intersection Theorem–Integral Identity Dictionary. P2D satisfies the u.c. in-tersection theorem C [A, l] ⇔ D satisfies the rational identity fC [A,l] = 0. Moreover, C [A, l] and fC

degenerate together.

Proof of “Moreover”. By the precursory observations above, the only inverted terms in f thatcannot be easily forced nonzero are the (m ·P )−1 from increment (v). For every set of substitutionsξ from D satisfying the other nondegeneracies to make one of these terms is undefined is for everyevaluation of C in P2D to degenerate; we retain the convenient property of Equivalence 4.1.

The inherent problem with these two dictionaries is evident:

fIeL=eR

is a (much) more complicated expression than the ei, and

IfC [A,l]

is a (much) more complicated configuration than C . We have equivalence relations on R[x] and onthe set of constructible intersection theorems, but no way to find nice representatives from either.This is why representatives like Amitsur’s S(p;n) are so important. If the answer to Question 4.6is “yes”, then we have such representatives for R[x]—for every intersection theorem C [A, l] therewill be p, n, and n-unreachable n such that C [A, l] holds in P2D ⇔ T (p; n,n) holds in D. Theidentities are manageable, if not concise or pretty; however, the corresponding intersection theoremrepresentatives IT (p;n,n) are obscenely complex structures.Note. While the explicit identities offered by Bergman and Rowen are indeed integral, recall fromthe note on page 34 that one frequently studies expressions with coefficients in some field F , whichare then evaluated at division rings containing F in their centers. Such expressions do not translateinto u.c. intersection theorems because only the points A0, A1, and A∞ are known to us on l1.

Conversely, the “rationalization” of the following section applies only to constructible intersec-tion theorems; in this sense integrality as a condition on expressions (and identities) is relatedto constructibility as a condition on configurations (and intersection theorems). Constructibilityis hardly a trivial property; the myriad of collinearities and concurrencies in the “mystic hexa-gram”, for example—each of which builds off an assumption of the previous, starting with Pap-pus’ Configuration—form a family of nonconstructible intersection theorems. Were both conditionslifted, would some larger equivalence theory gush forth? That is, what properties of objects in aprojective plane are revealed by a nonconstructible configuration which can be evaluated at them?

48

4.3 Storytelling

We have gone far in our goal of characterizing meso-desarguian–pappian intersection theorems,but in the process we seem to have lost hope of keeping a visual hold on most. In particular,intriguing intersection theorems are known to exist which evaluate and hold in P2Dp,n but evaluateand do not hold in each P2Dp,m with m - n, and a few are known explicitly. But what do theylook like, and how do they relate to one another? Interesting symmetries manifest in the confinedFano, Desargues, and Pappus Configurations, and such implications as S1[S, l1] ⇒ D [R, o] andthe division ring axioms from D [R, o] can be proven in pictures. Unfortunately for such purposes,however, the visible world is euclidean; the Theorems of Fano, Desargues, and Pappus are so-calledbecause they can be proven in euclidean space (which is why at least one “line” in the Fano Planeof Figure 1.3 must curve). Were we to construct in P2R the intersection theorem corresponding to,say, Bergman’s identity, the inverted terms [x, [x, a]−1] would all congregate at A∞ and absolutelynothing could be gathered from the degenerate mess. (In the base cases of F2 and S1 we gotlucky; it is not clear that even a rational identity of a minimal number of iterations will producean intersection theorem of minimal construction.)

Consider two options: We might take a minimal rational identity, construct its correspondingconfiguration, and break it into pieces which we could view separately in P2R. Alternatively, wemight look at “spatial” configurations evaluable in the projective 3-space P3D; these would involveplanes as well, but might require less algebra for an interpretation. Corollary 2.3 provides a generalsetting for a first step in both directions: Recall first that P3D contains such copies of P2D asthe planes [0 : 0 : 1 : 0] and [0 : 1 : 0 : 0], which share the “x-axis”. Label the points of a “pentacornof reference” E0 = (0 : 0 : 0 : 1), Ex = (1 : 0 : 0 : 0), Ey = (0 : 1 : 0 : 0), Ez = (0 : 0 : 1 : 0), andE∗ = (1 : 1 : 1 : 1). Label next the three coordinate planes Πij = E0EiEj and the celestial planeΠ∞ = ExEyEz.

E 0Ex

Ey

E z

Y·X

Y!

X 1

A 1X·Y

Y 1

X!

J

J'

I

I'

A

B

X

Y

Figure 4.2: X · Y and Y ·X on Separate Pages

Now consider Pappus’ Theorem itself—the intersection theorem holds in P2D precisely whenmultiplication in D is commutative. We can express this algebraic property more generally inP3D, “unfolding” Figure 3.3: First treat Πxz as a projective plane and set up l1 = E0Ex, A0 = E0,

49

A1 = (1 : 0 : 0 : 1), and A∞ = Ex. Construct X · Y using Ez as A′∞. (See figure 4.2.) Similarly,

construct Y · X in the plane Πxy with Ey as A′∞. The division ring D is commutative when for

every such pair X,Y the lines X∞Y1, Y∞X1, and l1 are concurrent.The intersection theorem is now spatial, and a bit more complex, but we can consider its com-

ponents separately—they lie on different “pages” bound together at l1. Specifically, an open book isa collection of planes, called pages, in 3-space through a common line, called the spine. (We could“close” this book to obtain the original intersection theorem.) Observe that the projection of this“spatial intersection theorem” is equivalent to Pappus’ Theorem, since we may use different pointsfor A in the constructions of X · Y and Y ·X.

Show Points

Figure 4.3: An Open Book

Construction 2. Let e : eL = eR be a rational identity in the indeterminates x1, . . . , xt, andlet {x1, . . . , xt, e1, . . . , en} be an incremental sequence for e. (Say en = eR and em = eL for somem ≤ n.) Take P3D, set l1 = A0A∞ = E0Ex, and set A1 = (1 :0 : 0 : 1). Construct a sequence Ce ofconfigurations by the following iterations, beginning with Ce = ∅:

(i) Construct Qref with points X1, . . . , Xt on ∆. Set the value s = 0.

(ii) Construct the point Ys+1 corresponding to the expression es+1 in the Xi. Continue construct-ing Ys+i until one of (es+k = 0), (es+k = 1), and (es+k = ej), for some j < s + k, is arational identity of F in the indeterminates x1, . . . , xt, . . . , es. (k will be > 1.) Append theconfiguration Cs+k−1 of constructions up to es+k−1 to Ce.

(iii) Construct Qref with points X1, . . . , Xt, . . . , Ys+k−1 on ∆. Reset the value s = s + k − 1 andgo to (ii).

The process of course ends when eL and eR are obtained; call the resulting sequence Ce astorybook for e. Then

Ce = (Ci1 , . . . ,Cim) with im = n.

Note that the storybook is heavily dependent upon the specific incremental sequence used to obtaine. By construction, each Cil can be evaluated in P2R isomorphically—that is, without overlap. Onthe other hand, one can evaluate the Cil on an open book in any P2D with spine l1 and assignmentsof A′

∞ on Πxy, as in Figure 4.2. There will be no shortage of pages from which to choose, since wepresume F to be an infinite field.

We could define the “thickness” of a rational identity e to be the minimum number of pagesrequired for a storybook to present it (ranging over the incremental sequences for e). This would

50

provide a gauge of complexity for e related to the largest n ∈ N for which division rings of p.i.degree n satisfy e. As we saw in the Pappus Configuration, the identity [x, y] = 0 has thickness 2.

Example. Wagner’s identity, written [x, y]2 · z = z · [x, y]2, requires the following pages:

page indeterminates constructions1 : {x, y, z} e4 = x · y2 : {x, y, z, e4} e5 = y · x, e6 = e4 − e5, e7 = e6

2, e8 = e7 · z3 : {x, y, z, e4, e5, e6, e7, e8} e9 = z · e7.

Since [x, y] requires two pages for its construction, and [x, y]2 · z and z · [x, y]2 must be constructedon different pages, Wager’s identity has thickness 3.

Recall that Desargues’ Theorem is clearly formulated in 3-space; it may turn out that spatialintersection theorems exist that are equivalent to such identities as B and the Tn,n and do notnecessarily degenerate for evaluations in P2D. Quite natural spatial configurations might “tell thestory” of e ∈ R[x] without bleeding into themselves.

Aside. We might expect the standard identity S2n to require, as does S2, a page for each summand∏2ni=1 xσ(i), and therefore (2n)! pages. This is certainly an upper bound, but can we improve it?For the time being, let us speak of “constructing” expressions by constructing the configurations

corresponding to them. We treat each Sn as the construction of two sums, which will coincidefor any evaluation in P2D when S2n holds in D. Provided the summands have been constructedon previous pages, only one page will be necessary for this operation; let us instead focus on thesummands.

Beginning with indeterminates x1, . . . , x2n, each of the constructions xi · xj with i < j can bedone on the first page, since there is always an evaluation of the xi in R such that no pair producethe same product. (The xj · xi cannot be constructed yet.) In fact, every product

∏i∈I xi over

subscript set I ⊆ [2n] in which the subscripts increase from left to right can be constructed in aconfiguration that will map isomorphically into P2R.

The second step will treat each of these products as a new indeterminate, so it makes senseto label them as such; xi · xj becomes xi,j , and so on until x[2n] = x1,...,2n. Thus we rememberwhich products (and in particular which summands) we have by their subscripts, though theyact like indeterminates in future steps (e.g. (x2 · x1) · x3 is distinct from x1,2 · x3). We don’tneed to associate the subscripts, since the summands

∏2ni=1 xσ(i) do not depend on the sequence

of multiplications that obtained them. (The specific configuration will, but Desargues’ Theorem,which implies associativity, tells us that the intersection theorems produced by such configurationstheorems will be equivalent.) Hence any product

∏i∈I xi ·

∏j∈J xj in which the indices increase

over the sets I and J (but not overall) can be constructed on the second page, provided the reverseproduct

∏j∈J xj ·

∏i∈I xi is not.

A better description of this phenomenon is in order. A word in the “letters” 1, 2, . . . , 2n ismerely a juxtaposition of finitely many letters. Under certain constraints on “steps”, our objectivein finding the thickness of S2n is to obtain all words of length [2n] without repeated letters in asfew steps as possible. The constraint is that, in any step with previously written words w1, . . . , wk,only one new word

∏i∈[k]wi may be written.

Since only one new word can be got from any two in a single step, we may describe the stepsin treelike fashion to get a diagram like Figure 4.4 for each step. Each step then adds another

51

illegal

x

y

z

xy

y·x

z·x

z·y

xy·z

z·x·y

y·x·z

yz

yz·x

Figure 4.4: A Word Tree for One Step

“dimension” to the tree as indeterminates are reassigned intermittently. Each vertex has “leftdegree” 2, and one of each pair of left edges is dominant, indicating the left factor of the product(an arbitrary choice between left and right).S2 requires two pages, while S4 can be (gruelingly) seen to require three. From the intuition

that one can effectively produce a power set’s count of new words from the old at each step, thefollowing hypothetical bound seems logical.

Conjecture 4.7. S2m has thickness ≤ m+ 1. More generally, S2n has thickness ≤ dlog2(2n)e+ 1.

C

As defined, the notion of thickness depends only on the characteristic and p.i. degree of R, andmight be called “(0, 1)-thickness” to bring attention to this. Were one so inclined, one might look at(p, n)-storybooks for expressions whose pages can be evaluated nondegenerately in Dp,n; to a singlerational expression would then correspond a “(p, n)-thickness” for each characteristic–p.i. degreepair, and an equivalence relation could even be imposed upon R[x] by

e1 ∼ e2 ⇔ ∀ p and n, e1 and e2 have the same (p, n)-thickness,

which would be a far cry from the equivalence on identities we’ve been using, given my mutualimplication.

52

A Coordinatizing the Desarguian Plane

Here we verify the remainder of the division ring axioms and conclude the proof of Equiva-lence 2.2. On page 23 we had established the ring operations + and · as group operations over∆ = l1 r {A∞} and ∆ r {A0}, respectively, with right distributivity to boot. Additive commuta-tivity, left distributivity, and the coordinatization of the entire plane remain.

Lemma A.1. Addition, defined as before, is commutative.

Proof. Look at the larger group PC[l∞,l∞] of collineations that fix no affine points (points off l∞),together with the identity. Let P and Q be any two affine points not collinear with A0 and defineτP ∈ PC[l∞,l∞] by τP (A0) = P and τQ similarly (a generalization of addition on ∆ to affine points).Consider PQ = τQ(P ). Since τQ fixes the line A0Q, its center must be Q′ = (A0Q)l∞. It mustthen take P to another point on the line (PQ′), so that PQ is collinear with P and Q′. But sinceτQ(A0) = Q and τQ fixes P ′ = (A0P )l∞, PQ must also be collinear with Q and P ′. Thus PQ is theintersection of PQ′ and QP ′, as pictured. By the same argument, so is QP = τP (Q). Since A0 wasreally just an arbitrary point off l∞, we actually have τP τQ = τQτP .

If we consider the collinear case of τR, where R is any

P'

Q'

l!

!P(Q)

A 0 P

Q

Figure A.1: Collineation τP (Q)

other affine point on A0P , and define −Q = τQ−1(A0),

we getτP τR = τP τQτQ

−1τR= (τP τQ)(τ−QτR)= (τQτP )(τRτ−Q)= τQ(τP τR)τQ−1

= τQτQ−1(τP τR) = τP τR,

since τP τR is a collineation that takes A0 to a point onPR, a line not incident with A0Q. Thus PC[l∞,l∞] is anabelian group. Since these collineations define additionon ∆, this addition is commutative.

Left distributivity is an even touchier condition, which will require the commutativity of PC[l∞,l∞].Retain the generalized notation τ = τP above for τ ∈ PC[Arl∞,l∞], τ(A0) = P .

Lemma A.2. Take P to be any affine point (other than A0) and set P∞ = (A0P )l∞. Then themap

ϕ : PC[A∞,l∞] → PC[P∞,l∞] defined by ϕ(τQ) = σQτPσQ−1

is a group homomorphism.

Proof. In the case P /∈ ∆, defineτ∗ = τA1

−1τP ,

so that τ∗(A1) = P . Say τ∗ ∈ PC[T∞,l∞]. For R,S ∈ ∆, then,

rclϕ(τR) = σRτRσR−1

= σR(τA1τ∗)σR−1

= (σRτA1σR−1)(σRτ∗σR−1)

= τR(σRτ∗σR−1),

53

so define τ∗R = σRτ∗σR

−1. Define τ∗S and τ∗R+S similarly, so that, by commutativity,

ϕ(τR)ϕ(τS) = (τRτ∗R)(τSτ∗S) = (τRτS)(τ∗Rτ∗S)

andϕ(τR+S) = τR+Sτ

∗R+S = (τRτS)τ∗R+S .

l!

P!

T!

A!

A 0

R'+S'

S'

R'

R SR+S

Figure A.2: PC[P∞,l∞] and PC[T∞,l∞]

Let R′ = ϕ(τR)(A0) and S′ and (R+ S)′ similarly. Note that

(R+ S)′ = τR+Sτ∗R+S(A0) = τ∗R+S(τR+S)(A0) = τ∗R+S(R+ S)

whileR′ + S′ := ϕ(τS(R′)) = ϕ(τR)ϕ(τS)(A0) = (τ∗Rτ

∗S)(τRτS)(A0) = (τ∗Rτ

∗S)(R+ S).

As images of A0 in PC[P∞,l∞], (R+S)′ and R′+S′ lie on A0P∞; as images of R+S in PC[T∞,l∞], bothalso lie on (R+S)T∞. These lines are distinct, so their intersection is unique; thus (R+S)′ = R′+S′

and ϕ(τR+S) = ϕ(τR)ϕ(τS).The case P ∈ ∆ is similar to the second case of Lemma A.1, and is left as an exercise.

For left distributivity, take ϕ in the second case of the lemma to send each τS to τP ·S = σSτPσS−1,

another member of PC[A∞,l∞]. The homomorphicity of ϕ then says

τP ·(Q+R) = ϕ(τQ+R) = ϕ(τQτR) != ϕ(τQ)ϕ(τR) = τP ·QτP ·R = τP ·Q+P ·R.

Apply the maps to A0 and we get the distributive law on ∆! We now have an algebraic structureon ∆, call it D, which we have just seen to be a division ring.

Designate l0 through A0, which will become the second

A'1

A!

A'!

A 0A 1

P'

A*

P

Figure A.3: The y-axis

axis, and label its points by

A′1 = (A∞A∗)l0P ′ = σP (A′

1).

(Remember that no point of l0 other than A0 and A′∞

can be fixed by a nontrivial σP .) Check that this agreeswith Figure A.3 by Lemma A.2, using PC[A′∞,l∞]. Theexisting assignment l0l∞ = A′

∞ completes the ′ theme.Since we’ve bijected l0 with l1, we have an additive group

isomorphism, which extends to a division ring isomorphism using the same σQ on both lines (inaccordance with the definitions).

54

Our intention is to coordinatize the entire plane using these two axes. To that end, let usintroduce the lowercase letters x, y, z to represent elements in D, while P , Q, etc. will representpoints in the plane. Thus if x is the element corresponding to P ∈ ∆, we assign P the coordinates(x : 0 : 1)—we also then take σx to mean σP . Similarly, we assign P ′ = σP (A′

1) the coordinates(0 :x :1). If P is off l1, l0, and l∞ then, using Figure A.4 as a guide, mark (A′

∞P )l1 = (x :0 :1) and(A∞P )l0 = (0 : y : 1). Assign P the coordinates (x : y : 1). These coordinates shall be homogeneousunder right multiplication by nonzero members of D. The scheme appropriately assigns A0 = (0 :0 :1), A1 = (1:0 :1), and A∗ = (1:1 :1).

We expect to assign coordinates to the points of l∞

(0:y:1)

(x:0:1)

A'1A!

A'!

A 0 A 1

P

Figure A.4: Coordinatization of P

by using the coordinates of points in their direction fromA0 and dropping the 1 from the third slot; for example,if A = (x :y : 1) then A′′

∞ = (A0A)l∞ should be (x :y : 0).However, to be sure that this assignment is well-defined,we must verify that lines are indeed defined by linearequations [a : b : c]. To this end, it will suffice to expressour homologies in terms of coordinates. (Since the oneswe’ve seen all fix l∞, we needn’t worry about coordinatesthere in the meantime.)

By Lemma 2.1 again we need only show that each

P!

(0:q':1)

(p':0:1)

(0:q:1)

(p:0:1)

A!

A'!

A 0

P

P'

Figure A.5: Homology ς(P )

homology ς ∈ PC[Arl∞,l∞] sends each point P = (p :q :1)to P ′ = (pλ : qλ : 1) for some λ ∈ D. The lemma assuresus that these multiples are all the points in the line A0Pbesides A0 and P∞ = (A0P )l∞. So picture P ′ = (p′ :q′ :1)on the line m = A0P . The perspectivity

[m]A′∞[ [l1]

takes P and P ′ to their shadows (p :0 :1) and (p′ :0 :1) onl1, while a perspectivity from A∞ takes them to (0:q :1)and (0:q′ :1) on l0. But since ς fixes l∞ pointwise it mustfix the two perspectors; thus ς(p : 0 : 1) = (p′ : 0 : 1) and ς(0 : q : 1) = (0 : q′ : 1). By the method ofthe lemma ς = σp−1p′ , and by that method and the way we defined coordinates on l0, ς = σq−1q′ . Afinal acknowledgment of uniqueness proves q−1q′ = p−1p′, thus λ = p−1p′ satisfies our criterion.

Thus if P = (p1 : p2 : 1) and Q = (q1 : q2 : 1) define the same point at infinity P∞—that is, ifP , Q, and A0 are collinear—then there is some λ ∈ D such that qi = piλ, so that (p1 : p2 : 0) and(q1 :q2 : 0) are the same set of homogeneous coordinates. Either (indeed, both) may be assigned toP∞. We conclude, gratefully, that P = P2D.

55

B Permuting Perspectors: S3 on R[x]

We wrap up the paper with a delightful and superfluous bow: an application of the automorphismgroup Aut P2D to the collection of rational identities of D. As we’ve seen, the cataloguing ofrational identities holding in certain division rings has been a noble and fruitful venture. We’ve alsobecome aware that fairly simple intersection theorems can produce wildly complicated identities. ByProposition 3.4, to prove an intersection theorem in P2D it suffices to verify it on the quandrangle ofreference. (Hua’s identity, for instance, produces a universal intersection theorem via the operationsof Section 2.3. This intersection theorem must then hold in any desarguian projective plane.) Whenwe stick to the points and operations on l1 the rational expressions are greatly simplified.

A'!

Z –1

–ZZ+WZ·W

m

A 0

A!

A

I

I'

l!

l 1

l'l

A 1

l''

J

WZ

Figure B.1: The Original Setup & Projectivities

Notice (Figure B.1) that three distinct perspectors—A, I, and I ′—direct the projectivities whichdefine the operations on ∆. The sides of triangle AII ′ intersect l1 at the key points A0, A1, andA∞. What, then, would happen to the rational expressions corresponding to our constructions ifwe were to build them after a permutation of this triangle? We shall see, all the while preservingthe canvas l1.

There is a transformation in PGL3(D) sending any quadrangle to any other, so take [A] ∈PGL3(D) which sends

A∞ 7→ A0 A1 7→ A1 A 7→ I ′ I 7→ I;

it follows that [A](A0) = A∞ and [A](I ′) = A as corresponding diagonal points of the quadranglesA∞A1AI and A0A1I

′I. We can find a representation of the matrix A by looking at coordinates:

[A](1 :0 :0) = (0:0 :1)[A](1 :0 :1) = (1:0 :1)[A](a :b :1) = (a− 1:b :0)[A](a :b :0) = (a :b :0),

so, solving the system of equations, one form A might take is

A =

0 −aαb−1 α0 −bαb−1 0α −αab−1 0

,

56

where α ∈ D is arbitrary.

X –1

–X(Y –1+X –1)–1Y·X

mA 0

A!

A

I

I'

l!

l 1

l'

l

A 1

l''

J

Y

X

Figure B.2: Enter A

Now consider X and Y on l1, the images of Z and W under [A] (Figure B.2). These are stillarbitrary points on l1, provided neither Z nor W is A0; but the constructions in Figure B.1 do notperform the same operations on D as before. Look at −Z. Following coordinates, since

−Z = (IZ ′)l1 = (I((A0Z∞)l))l1 = (I((A0((AZ)l∞))l))l1,

we getAZ 7→ I ′X = (x :0 :1)(a− 1:b :0) = [1:(a− 1)b−1 :−x]Z∞ 7→ (I ′X)l′ = [1:(a− 1)b−1 :−x][−a−1 :b−1 :0] = (a :b :x−1)

=: X∗∞

A0Z∞ 7→ A∞X∗∞ = (1:0 :0)(a :b :x−1) = [0:1 :−bx]

Z ′ 7→ (A∞X∗∞)m = [0:1 :−bx][b :1− bab−1 :0] = ((a− 1)x :bx :1)

=: X ′∗

IZ ′ 7→ IX ′∗ = (a :b :0)((a− 1)x :bx :1) = [1:−ab−1 :x]−Z 7→ (IX ′∗)l1 = [1:−ab−1 :x][−x :0 :1]

= −X,

and negation −X is preserved as an operation. It turns out that X−1 is also preserved. However,since l∞ and l′ are swapped as well as A and I ′, we get

Z∞ 7→ X1 W1 7→ Y∞,

henceZ ·W 7→ Y ·X;

multiplication is reversed! If we go through the motions again we find that

Z +W 7→ (Y −1 +X−1)−1,

and via additive commutativity this translates to the lovely

(Z +W )−1 7→ X−1 + Y −1.

57

Now for the interpretation in rational identities: Let e(x1, . . . , xt) be a rational expression int indeterminates. From e obtain the expression e[A] as follows: Let e1, . . . , en be an incrementalsequence for e. For each 1 ≤ k ≤ n,

if ek =

−eiei + ejei−1

ei · ej

then set e′k =

−e′i(e′i

−1 + e′j−1)−1

e′i−1

e′j · e′i

.

Then D satisfies the rational identity e ⇔ P2D satisfies the intersection theorem IeProp. 3.4⇐⇒ P2D

satisfies the intersection theorem [A](Ie) ⇔ D satisfies the rational identity e[A].

Example. Recall Hua’s identity

(x−1 + (y−1 − x)−1)−1 = x− x · y · x,

which holds in every division ring. We now have the distinct identity

(((x−1)−1 + ((y−1 − x)−1)−1)−1)−1 = (x−1 − (x · y · x)−1)−1,

which also holds in every division ring. We can reduce this to the identity

x+ (y−1 − x) = (x−1 − x−1 · y−1 · x−1)−1

provided this one still holds when y−1 = x—we want this identity to be 0 or undefined where Hua’sis undefined. (It is quick and easy to check that this is still degenerate.)

Note. We must be careful; this is treacherous ground. Working directly from the points, the identityx·y = y·x pleasantly becomes y·x = x·y, but x·y−y·x = 0 becomes ((y·x)−1−(x·y)−1)−1 = ∞, since[A](A0) = A∞—a degenerate rational identity equivalent to the nondegenerate (y ·x)−1−(x·y)−1 =0. To avoid such cases, it will be helpful to write identities as equalities of expressions in at leastone xi.

The collineation [A] ∈ PGL3D permutes the points {A0, A1, A∞} on l1 and the vertices oftriangle AII ′. It is plain to see that there are other collineations that round out a group, isomorphicto S3 by its actions on these points. Let’s call this known matrix

A(0 ∞) =

0 −aαb−1 α0 −bαb−1 0α −αab−1 0

,

interpreting S3 as the group of permutations of {0, 1,∞}. It is a straightforward check that thematrix

A(1 ∞) =

β (aβ − βa)b−1 00 bβb−1 0β β(1− a)b−1 −β

induces a collineation that takes

A∞ 7→ A1 A0 7→ A0 A 7→ I I ′ 7→ I ′,

58

where again the nonzero element β is arbitrary.These two transpositions generate a representation of S3 in GL3D:

A(0 1) = A(0 ∞)A(0 1)A(0 ∞)

A(0 1 ∞) = A(0 ∞)A(0 1)

A(0 ∞ 1) = A(0 1)A(0 ∞),

and of course A1 = I. Each Aσ corresponds to a rewording eσ of identities e, such that each eσ

holds in exactly those division rings that satisfy e.Fleshing out these products, we get

A(0 1) =

−γ (aγ − γ − γa)b−1 γ0 bγb−1 00 0 γ

,

A(0 1 ∞) =

0 −aµb−1 µ0 −bµb−1 0−µ µ(a− 1)b−1 µ

,

and

A(0 ∞ 1) =

ν (−aν + ν − νa)b−1 −ν0 −bνb−1 0ν −νab−1 0

,

where γ, µ, and ν are arbitrary in D×. Observe that, on l1, the arbitrary values serve only toconjugate elements of D; for instance,

A(0 1) · (z :0 :1) = (−γz + γ :0 :γ) = (−zγ + 1:0:1) = ((−z + 1)γ :0 :1).

This is an automorphism; for simplicity, let us assume that each of these values is 1 (and leave thereduced forms of the matrices above to the reader).

Once we have these matrices it is easy to see how each collineation affects the operations −, −1,+, and ·. Looking again at [A(0 1)], using the changes of variables x = −z+1 and y = −w+1, andapplying some arithmetic rules which apply in all division rings,

[A(0 1)](z :0 :1) = (−z + 1:0:1) = (x :0 :1)[A(0 1)](−z :0 :1) = (z + 1:0:1) = (−x+ 2:0:1)[A(0 1)](z−1 :0 :1) = (−z−1 + 1:0:1) = (1− (1− x)−1 :0 :1)[A(0 1)](z + w :0 :1) = (−(z + w) + 1:0 :1) = ((x+ y)− 1:0 :1)[A(0 1)](z · w :0 :1) = (−z · w + 1:0:1) = ((x+ y)− x · y :0 :1).

(We understand 2 to mean (1+1)). These are times when the hierarchy imposed on the expressionsin R[x] comes heavily into play. The simple identity

(z + w) + (−(z + w)) = 0,

which we habitually shorten to (z+w)− (z+w) = 0, can no longer endure such flights of commonsense. Piece by piece, we translate it:

z + w 7→ (x+ y)− 1−(z + w) 7→ −((x+ y)− 1) + 2

(z + w) + (−(z + w)) 7→ ((x+ y)− 1) + (−((x+ y)− 1) + 2)− 10 7→ 1;

59

thus we get the rational identity

((x+ y)− 1) + (−((x+ y)− 1) + 2)− 1 = 1,

which indeed holds via properties of addition.

Table B.1: Permuted Operations

[A1] [A(0 1)] [A(0 ∞)] [A(1 ∞)]z − z + 1 z−1 z · (z − 1)−1

= (1− z−1)−1

= x = x = x

−z z + 1 − z−1 z · (z + 1)−1

= (1 + z−1)−1

= −x+ 2 = −x = (−x+ 2)−1

z−1 − z−1 + 1 z z−1 · (z−1 − 1)−1

= (1− z)−1

= (x− 1)−1 + 1 = x−1 = ((x−1 − 1)−1 + 1)−1

z + w 1− (z + w) (z + w)−1 (z + w)((z + w)− 1)−1

= (1− (z + w)−1)−1

= (x+ y)− 1 = (x−1 + y−1)−1 = (1− (2− (x−1 + y−1))−1)−1

z · w 1− z · w (z · w)−1 (z · w) · (z · w − 1)−1

= w−1 · z−1 = (1− w−1 · z−1)−1

= (x+ y)− x · y = y · x = x · ((x+ y)− 1)−1 · y

Following suit, we can permute R[x] under the other [Aσ] by composing the transformationsabove, and the appropriate changes in variables for each may thence be derived. The literal trans-formations (still in terms of z and w) are listed in Table B.1, along with the expressions they yieldafter the reassignments x = [Aσ]z and y = [Aσ]w and applying some simplifications allowed by thedivision ring axioms and the distinctions z, w 6= 0, 1 (for instance, (z−1)−1 = z)—simplificationsthat will not affect whether an identity holds in a given D.

60

Bibliography

[1] S. A. Amitsur; “Rational Identities and Applications To Algebra and Geometry”, Journal ofAlgebra 3 (1966), 304–359.

[2] S. A. Amitsur; “Polynomial Identities”, Israel Journal of Mathematics 19 (1974), 183–199.

[3] G. M. Bergman; “Rational Relations and Rational Identities in Division Rings I” & “II”,Journal of Algebra 43 (1976), 252–266.

[4] L. Le Bruyn; “Rational Identities of Matrices Revisited”, Department Wiskunde en Informat-ica: January 10, 1991.

[5] P. J. Cameron; Projective and Polar Spaces, Second Edition.http://www.maths.qmul.ac.uk/∼pjc/pps/, 14 September 2000.

[6] H. S. M. Coxeter; Projective Geometry, Second Edition. Springer: New York, 1987.

[7] M. Dehn; “Uber die Grundlagen der projektiven Geometrie und allgemeine Zahlsysteme”,Mathematische Annalen 85 (1922), 184–194.

[8] R. Hartshorne; Foundations of Projective Geometry. Benjamin Press: Cambridge, 1967.

[9] M. Kubovy, C. Tyler; The Arrow In the Eye: The Psychology of Perspective and RenaissanceArt. WebExhibits: http://webexhibits.org/arrowintheeye/, 2005.

[10] L. H. Rowen; Polynomial Identities In Ring Theory. Academic Press: New York, 1980.

[11] H. Salzmann, D. Betten, T. Grundhofer, H. Hahl, R. Lowen, M. Stroppel; Compact ProjectivePlanes. Walter de Gruyter: Berlin / New York, 1995.

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Vita

Jason Cory Brunson was born, fed, sheltered, and raised by his parents Barry Brunson andPansy Waycaster and assisted in mischief by his sister Nicole, of each of whom much more shouldbe known, in particular that they remain supportive and proud of every honest endeavor he makes.He joined Virginia Tech as an undergraduate in August of 2000 and earned Bachelor’s Degrees inMathematics and Statistics in 2004, the former as part of a 5-year Master’s program in the MathDepartment, which culminated in this document. During his career at VT, which will continue atleast another year in the Ph.D. program, he has served in the Math and Stat Clubs, performedwith the Tolls of Madness Cast, and joined the Executive Board of the Student Chapter of SIAM.He hopes ultimately to double the length of this vita.

Typeset in LATEXwith TEXShop (v1.35e).Figures, except 1.6, created with The Geometer’s SketchPad (v4.06).Email jabrunso@math.vt.edu for source files.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License.To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/2.0/ orsend a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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