ordered field

Ordered eldFrom Wikipedia, the free encyclopedia

Contents

1 Finite eld 11.1 Denitions, rst examples, and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Explicit construction of nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Non-prime elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 GF(p2) for an odd prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 GF(8) and GF(27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.5 GF(16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 Discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Frobenius automorphism and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Polynomial factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.1 Irreducible polynomials of a given degree . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6.2 Number of monic irreducible polynomials of a given degree over a nite eld . . . . . . . . 8

1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8.2 Wedderburns little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Glossary of eld theory 112.1 Denition of a eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Types of elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

i

ii CONTENTS

2.7 Extensions of Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Ordered eld 173.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.2 Positive cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Equivalence of the two denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.4 Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Properties of ordered elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1 Vector spaces over an ordered eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Examples of ordered elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Which elds can be ordered? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Topology induced by the order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Harrison topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Superordered elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Perfect eld 224.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Field extension over a perfect eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Perfect closure and perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Separable extension 255.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 275.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.11 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CONTENTS iii

5.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 1

Finite eld

In mathematics, a nite eld or Galois eld (so-named in honor of variste Galois) is a eld that contains a nitenumber of elements. As with any eld, a nite eld is a set on which the operations of multiplication, addition,subtraction and division are dened and satisfy certain basic rules. The most common examples of nite elds aregiven by the integers mod n when n is a prime number.The number of elements of a nite eld is called its order. A nite eld of order q exists if and only if the order q isa prime power pk (where p is a prime number and k is a positive integer). All elds of a given order are isomorphic.In a eld of order pk, adding p copies of any element always results in zero; that is, the characteristic of the eld is p.In a nite eld of order q, the polynomial Xq X has all q elements of the nite eld as roots. The non-zero elementsof a nite eld form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powersof a single element called a primitive element of the eld (in general there will be several primitive elements for agiven eld.)A eld has, by denition, a commutative multiplication operation. A more general algebraic structure that satisesall the other axioms of a eld but isn't required to have a commutative multiplication is called a division ring (orsometimes skeweld). A nite division ring is a nite eld by Wedderburns little theorem. This result shows that theniteness condition in the denition of a nite eld can have algebraic consequences.Finite elds are fundamental in a number of areas of mathematics and computer science, including number theory,algebraic geometry, Galois theory, nite geometry, cryptography and coding theory.

Commutative rings integral domains integrally closed domains unique factorization do-mains principal ideal domains Euclidean domains elds nite elds

1.1 Denitions, rst examples, and basic propertiesA nite eld is a nite set on which the four operations multiplication, addition, subtraction and division (excludingby zero) are dened, satisfying the rules of arithmetic known as the eld axioms. The simplest examples of niteelds are the prime elds: for each prime number p, the eld GF(p) (also denoted Z/pZ, Fp , or Fp) of order (that is,size) p is easily constructed as the integers modulo p.The elements of a prime eld may be represented by integers in the range 0, ..., p 1. The sum, the dierence andthe product are computed by taking the remainder by p of the integer result. The multiplicative inverse of an elementmay be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm Modular integers).Let F be a nite eld. For any element x in F and any integer n, let us denote by nx the sum of n copies of x. Theleast positive n such that n1 = 0 must exist and is prime; it is called the characteristic of the eld.If the characteristic of F is p, the operation (k; x) 7! k xmakes F a GF(p)-vector space. It follows that the numberof elements of F is pn.For every prime number p and every positive integer n, there are nite elds of order pn, and all these elds areisomorphic (see Existence and uniqueness below). One may therefore identify all elds of order pn, which aretherefore unambiguously denoted Fpn , Fpn or GF(pn), where the letters GF stand for Galois eld.[1]

1

2 CHAPTER 1. FINITE FIELD

The identity

(x+ y)p = xp + yp

is true (for every x and y) in a eld of characteristic p. (This follows from the fact that all, except the rst and the last,binomial coecients of the expansion of (x+ y)p are multiples of p).For every element x in the prime eld GF(p), one has xp = x (This is an immediate consequence of Fermats littletheorem, and this may be easily proved as follows: the equality is trivially true for x = 0 and x = 1; one obtains theresult for the other elements of GF(p) by applying the above identity to x and 1, where x successively takes the values1, 2, ..., p 1 modulo p.) This implies the equality

Xp X =Y

a2GF(p)(X a)

for polynomials over GF(p). More generally, every element in GF(pn) satises the polynomial equation xpn x = 0.Any nite eld extension of a nite eld is separable and simple. That is, if E is a nite eld and F is a subeld ofE, then E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon,nite elds are perfect.

1.2 Existence and uniquenessLet q = pn be a prime power, and F be the splitting eld of the polynomial

P = Xq Xover the prime eld GF(p). This means that F is a nite eld of lowest order, in which P has q distinct roots (theroots are distinct, as the formal derivative of P is equal to 1). Above identity shows that the sum and the product oftwo roots of P are roots of P, as well as the multiplicative inverse of a root of P. In other word, the roots of P form aeld of order q, which is equal to F by the minimality of the splitting eld.The uniqueness up to isomorphism of splitting elds implies thus that all elds of order q are isomorphic.In summary, we have the following classication theorem rst proved in 1893 by E. H. Moore:[2]

The order of a nite eld is a prime power. For every prime power q there are elds of orderq, and they are all isomorphic. In these elds, every element satises

xq = x;

and the polynomial Xq X factors as

Xq X =Ya2F

(X a):

It follows that GF(pn) contains a subeld isomorphic to GF(pm) if and only if m is a divisor of n; in that case, thissubeld is unique. In fact, the polynomial Xpm X divides Xpn X if and only if m is a divisor of n.

1.3 Explicit construction of nite elds

1.3.1 Non-prime eldsGiven a prime power q = pn with p prime and n > 1, the eld GF(q) may be explicitly constructed in the followingway. One chooses rst an irreducible polynomial P in GF(p)[X] of degree n (such an irreducible polynomial alwaysexists). Then the quotient ring

1.3. EXPLICIT CONSTRUCTION OF FINITE FIELDS 3

GF(q) = GF(p)[X]/(P )

of the polynomial ring GF(p)[X] by the ideal generated by P is a eld of order q.More explicitly, the elements of GF(q) are the polynomials over GF(p) whose degree is strictly less than n. Theaddition and the subtraction are those of polynomials over GF(p). The product of two elements is the remainderof the Euclidean division by P of the product in GF(p)[X]. The multiplicative inverse of a non-zero element maybe computed with the extended Euclidean algorithm; see Extended Euclidean algorithm Simple algebraic eldextensions.Except in the construction of GF(4), there are several possible choices for P, which produce isomorphic results. Tosimplify the Euclidean division, for P one commonly chooses polynomials of the form

Xn + aX + b;

which make the needed Euclidean divisions very ecient. However, for some elds, typically in characteristic 2,irreducible polynomials of the formXn+ aX + bmay not exist. In characteristic 2, if the polynomial Xn + X + 1 isreducible, it is recommended to choose Xn + Xk + 1 with the lowest possible k that makes the polynomial irreducible.If all these trinomials are reducible, one chooses pentanomials Xn + Xa + Xb + Xc + 1, as polynomials of degreegreater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root.[3]

In the next sections, we will show how this general construction method works for small nite elds.

1.3.2 Field with four elementsOver GF(2), there is only one irreducible polynomial of degree 2:

X2 +X + 1

Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and

GF(4) = GF(2)[X]/(X2 +X + 1):

If one denotes a a root of this polynomial in GF(4), the tables of the operations in GF(4) are the following. Thereis no table for subtraction, as, in every eld of characteristic 2, subtraction is identical to addition. In the third table,for the division of x by y, x must be read on the left, and y on the top.

1.3.3 GF(p2) for an odd prime pFor applying above general construction of nite elds in the case of GF(p2), one has to nd an irreducible polynomialof degree 2. For p = 2, this has been done in the preceding section. If p is an odd prime, there are always irreduciblepolynomials of the form X2 r, with r in GF(p).More precisely, the polynomial X2 r is irreducible over GF(p) if and only if r is a quadratic non-residue modulop (this is almost the denition of a quadratic non-residue). There are p12 quadratic non-residues modulo p. Forexample, 2 is a quadratic non-residue for p = 3, 5, 11, 13, ..., and 3 is a quadratic non-residue for p = 5, 7, 17, .... Ifp 3 mod 4, that is p = 3, 7, 11, 19, ..., one may choose 1 p 1 as a quadratic non-residue, which allows us tohave a very simple irreducible polynomial X2 + 1.Having chosen a quadratic non-residue r, let be a symbolic square root of r, that is a symbol which has the property2 = r, in the same way as the complex number i is a symbolic square root of 1. Then, the elements of GF(p2) areall the linear expressions

a+ b;


with a and b in GF(p). The operations on GF(p2) are dened as follows (the operations between elements of GF(p)represented by Latin letters are the operations in GF(p)):

(a+ b) = a+ (b)(a+ b) + (c+ d) = (a+ c) + (b+ d)

(a+ b)(c+ d) = (ac+ rbd) + (ad+ bc)

(a+ b)1 = a(a2 rb2)1 + (b)(a2 rb2)1

1.3.4 GF(8) and GF(27)The polynomial

X3 X 1is irreducible over GF(2) and GF(3), that is, it is irreducible modulo 2 and 3 (to show this it suce to show that it hasno root in GF(2) nor in GF(3)). It follows that the elements of GF(8) and GF(27) may be represented by expressions

a+ b+ c2;

where a, b, c are elements of GF(2) or GF(3) (respectively), and is a symbol such that

3 = + 1:

The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be dened as follows; in followingformulas, the operations between elements of GF(2) or GF(3), represented by Latin letters are the operations in GF(2)or GF(3), respectively:

(a+ b+ c2) = a+ (b)+ (c)2 (ForGF (8); identity) the is operation this(a+ b+ c2) + (d+ e+ f2) = (a+ d) + (b+ e)+ (c+ f)2

(a+ b+ c2)(d+ e+ f2) = (ad+ bf + ce) + (ae+ bd+ bf + ce+ cf)+ (af + be+ cd+ cf)2

1.3.5 GF(16)The polynomial

X4 +X + 1

is irreducible over GF(2), that is, it is irreducible modulo 2. It follows that the elements of GF(16) may be representedby expressions

a+ b+ c2 + d3;

where a, b, c, d are either 0 or 1 (elements of GF(2)), and is a symbol such that

4 = + 1:

As the characteristic of GF(2) is 2, each element is its additive inverse in GF(16). The addition and multiplicationon GF(16) may be dened as follows; in following formulas, the operations between elements of GF(2), representedby Latin letters are the operations in GF(2).

1.4. MULTIPLICATIVE STRUCTURE 5

(a+ b+ c2 + d3) + (e+ f+ g2 + h3) = (a+ e) + (b+ f)+ (c+ g)2 + (d+ h)3

(a+ b+ c2 + d3)(e+ f+ g2 + h3) = (ae+ bh+ cg + df) + (af + be+ bh+ cg + df + ch+ dg) +

(ag + bf + ce+ ch+ dg + dh)2 + (ah+ bg + cf + de+ dh)3

1.4 Multiplicative structureThe set of non-zero elements in GF(q) is an Abelian group under the multiplication, of order q 1. By Lagrangestheorem, there exists a divisor k of q 1 such that xk = 1 for every non-zero x in GF(q). As the equation Xk = 1 has atmost k solutions in any eld, q 1 is the lowest possible value for k. The structure theorem of nite Abelian groupsimplies that this multiplicative group is cyclic, that all non-zero elements are powers of single element. In summary:

The multiplicative group of the non-zero elements in GF(q) is cyclic, and there exist an element a, suchthat the q 1 non-zero elements of GF(q) are a, a2, ..., aq2, aq1 = 1.

Such an element a is called a primitive element. Unless q = 2, 3, the primitive element is not unique. The number ofprimitive elements is (q 1) where is Eulers totient function.Above result implies that xq = x for every x in GF(q). The particular case where q is prime is Fermats little theorem.

1.4.1 Discrete logarithmIf a is a primitive element in GF(q), then for any non-zero element x in F, there is a unique integer n with 0 n q 2 such that

x = an.

This integer n is called the discrete logarithm of x to the base a.While the computation of an is rather easy, by using, for example, exponentiation by squaring, the reciprocal oper-ation, the computation of the discrete logarithm is dicult. This has been used in various cryptographic protocols,see Discrete logarithm for details.When the nonzero elements of GF(q) are represented by their discrete logarithms, multiplication and division areeasy, as they reduce to addition and subtraction modulo q 1. However, addition amounts to computing the discretelogarithm of am + an. The identity

am + an = an(amn + 1)

allows one to solve this problem by constructing the table of the discrete logarithms of an + 1, called Zechs logarithms,for n = 0, ..., q 2 (it is convenient to dene the discrete logarithm of zero as being ).Zechs logarithms are useful for large computations, such as linear algebra over medium-sized elds, that is, elds thatare suciently large for making natural algorithms inecient, but not too large, as one has to pre-compute a table ofthe same size as the order of the eld.

1.4.2 Roots of unityEvery nonzero element of a nite eld is a root of unity, as xq1 = 1 for every nonzero element of GF(q).If n is a positive integer, a nth primitive root of unity is a solution of the equation xn = 1 that is not a solution of theequation xm = 1 for any positive integer m < n. If a is a nth primitive root of unity in a eld F, then F contains all then roots of unity, which are 1, a, a2, ..., an1.The eld GF(q) contains a nth primitive root of unity if and only if n is a divisor of q 1; if n is a divisor of q 1,then the number of primitive nth roots of unity in GF(q) is (n) (Eulers totient function). The number of nth rootsof unity in GF(q) is gcd(n, q 1).


In a eld of characteristic p, every (np)th root of unity is also a nth root of unity. It follows that primitive (np)th rootsof unity never exist in a eld of characteristic p.On the other hand, if n is coprime to p, the roots of the nth cyclotomic polynomial are distinct in every eld ofcharacteristic p, as this polynomial is a divisor of Xn 1, which has 1 as formal derivative. It follows that the nthcyclotomic polynomial factors over GF(p) into distinct irreducible polynomials that have all the same degree, say d,and that GF(pd) is the smallest eld of characteristic p that contains the nth primitive roots of unity.

1.4.3 Example

The eld GF(64) has several interesting properties that smaller elds do not share. Specically, it has two subeldssuch that neither is a subeld of the other, not all generators (elements having a minimal polynomial of degree 6 overGF(2)) are primitive elements, and the primitive elements are not all conjugate under the Galois group.The order of this eld being 26, and the divisors of 6 being 1, 2, 3, 6, the subelds of GF(64) are GF(2), GF(22) =GF(4), GF(23) = GF(8), and GF(64) itself. As 2 and 3 are coprime, the intersection of GF(4) and GF(8) in GF(64)is the prime eld GF(2).The union of GF(4) and GF(8) has thus 10 elements. The remaining 54 elements of GF(64) generate GF(64) in thesense that no other subeld contains any of them. It follows that they are roots of irreducible polynomials of degree6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54/6 irreducible monic polynomials of degree 6.This may be veried by factoring X64 X over GF(2).The elements of GF(64) are primitive nth roots of unity for some n dividing 63. As the 3rd and the 7th roots of unitybelong to GF(4) and GF(8), respectively, the 54 generators are primitive nth roots of unity for some n in {9, 21, 63}.Eulers totient function shows that there are 6 primitive 9th roots of unity, 12 primitive 21st roots of unity, and 36primitive 63rd roots of unity. Summing these numbers, one nds again 54 elements.By factoring the cyclotomic polynomials over GF(2), one nds that:

The six primitive 9th roots of unity are roots of

X6 +X3 + 1;

and are all conjugate under the action of the Galois group.

The twelve primitive 21st roots of unity are roots of

(X6 +X4 +X2 +X + 1)(X6 +X5 +X4 +X2 + 1):

They form two orbits under the action of the Galois group. As the two factors are reciprocal to eachother, a root and its (multiplicative) inverse do not belong to the same orbit.

The 36 primitive elements of GF(64) are the roots of

(X6+X4+X3+X+1)(X6+X+1)(X6+X5+1)(X6+X5+X3+X2+1)(X6+X5+X2+X+1)(X6+X5+X4+X+1);

They split into 6 orbits of 6 elements under the action of the Galois group.

This shows that the best choice to construct GF(64) is to dene it as GF(2)[X]/(X6 + X + 1). In fact, this generatoris a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.

1.5. FROBENIUS AUTOMORPHISM AND GALOIS THEORY 7

1.5 Frobenius automorphism and Galois theoryIn this section, p is a prime number, and q = pn is a power of p.In GF(q), the identity (x+ y)p = xp + yp implies that the map

' : x 7! xp

is a GF(p)-linear endomorphism and a eld automorphism of GF(q), which xes every element of the subeld GF(p).It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.Denoting by 'k the composition of ' with itself, k times, we have

'k : x 7! xpk :

It has been shown in the preceding section that 'n is the identity. For 0 < k < n, the automorphism 'k is not theidentity, as, otherwise, the polynomial

Xpk X

would have more than pk roots.There are no other GF(p)-automorphisms of GF(q). In other words, GF(pn) has exactly n GF(p)-automorphisms,which are

Id = '0; '; '2; : : : ; 'n1:

In terms of Galois theory, this means that GF(pn) is a Galois extension of GF(p), which has a cyclic Galois group.The fact that the Frobenius map is surjective implies that every nite eld is perfect.

1.6 Polynomial factorizationMain article: Factorization of polynomials over nite elds

If F is a nite eld, a non-constant monic polynomial with coecients in F is irreducible over F, if it is not the productof two non-constant monic polynomials, with coecients in F.As every polynomial ring over a eld is a unique factorization domain, every monic polynomial over a nite eld maybe factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.There are ecient algorithms for testing polynomial irreducibility and factoring polynomials over nite eld. Theyare a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, everycomputer algebra system has functions for factoring polynomials over nite elds, or, at least, over nite prime elds.

1.6.1 Irreducible polynomials of a given degreeThe polynomial

Xq X

factors into linear factors over a eld of order q. More precisely, this polynomial is the product of all monic polyno-mials of degree one over a eld of order q.


This implies that, if q = pn that Xq X is the product of all monic irreducible polynomials over GF(p), whose degreedivides n. In fact, if P is an irreducible factor over GF(p) of Xq X, its degree divides n, as its splitting eld iscontained in GF(pn). Conversely, if P is an irreducible monic polynomial over GF(p) of degree d dividing n, itdenes a eld extension of degree d, which is contained in GF(pn), and all roots of P belong to GF(pn), and are rootsof Xq X; thus P divides Xq X. As Xq X does not have any multiple factor, it is thus the product of all theirreducible monic polynomials that divide it.This property is used to compute the product of the irreducible factors of each degree of polynomials over GF(p);see Distinct degree factorization.

1.6.2 Number of monic irreducible polynomials of a given degree over a nite eldThe number N(q,n) of monic irreducible polynomials of degree n over GF(q) is given by[4]

N(q; n) =1

n

Xdjn

(d)qnd ;

where is the Mbius function. This formula is almost a direct consequence of above property of Xq X.By the above formula, the number of irreducible (not necessarily monic) polynomials of degree n over GF(q) is (q 1)N(q, n).A (slightly simpler) lower bound for N(q, n) is

N(q; n) 1n

0@qn Xpjn; pprime

qnp

1A :One may easily deduce that, for every q and every n, there is at least one irreducible polynomial of degree n overGF(q). This lower bound is sharp for q = n = 2.

1.7 ApplicationsIn cryptography, the diculty of the discrete logarithm problem in nite elds or in elliptic curves is the basis ofseveral widely used protocols, such as the DieHellman protocol. For example, in 2014, the secure connection toWikipedia involves the elliptic curve DieHellman protocol (ECDHE) over a large nite eld.[5] In coding theory,many codes are constructed as subspaces of vector spaces over nite elds.Finite elds are widely used in number theory, as many problems over the integers may be solved by reducing themmodulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization andlinear algebra over the eld of rational numbers proceed by reduction modulo one or several primes, and then recon-struction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.Similarly many theoretical problems in number theory can be solved by considering their reductions modulo someor all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry weremotivated by the need to enlarge the power of these modular methods. Wiles proof of Fermats Last Theorem is anexample of a deep result involving many mathematical tools, including nite elds.

1.8 Extensions

1.8.1 Algebraic closureA nite eld F is not algebraically closed. To demonstrate this, consider the polynomial

f(T ) = 1 +Y2F

(T );

1.9. SEE ALSO 9

which has no roots in F, since f () = 1 for all in F.The direct limit of the system:

{Fp, Fp2, ..., Fpn, ...},

with inclusion, is an innite eld. It is the algebraic closure of all the elds in the system, and is denoted by: Fp .The inclusions commute with the Frobenius map, as it is dened the same way on each eld (x x p ), so theFrobenius map denes an automorphism of Fp , which carries all subelds back to themselves. In fact Fpn can berecovered as the xed points of the nth iterate of the Frobenius map.However unlike the case of nite elds, the Frobenius automorphism on Fp has innite order, and it does not generatethe full group of automorphisms of this eld. That is, there are automorphisms of Fp which are not a power of theFrobenius map. However, the group generated by the Frobenius map is a dense subgroup of the automorphism groupin the Krull topology. Algebraically, this corresponds to the additive group Z being dense in the pronite integers(direct product of the p-adic integers over all primes p, with the product topology).If we actually construct our nite elds in such a fashion that Fpn is contained in Fpm whenever n divides m, then thisdirect limit can be constructed as the union of all these elds. Even if we do not construct our elds this way, we canstill speak of the algebraic closure, but some more delicacy is required in its construction.

1.8.2 Wedderburns little theoremA division ring is a generalization of eld. Division rings are not assumed to be commutative. There are no non-commutative nite division rings: Wedderburns little theorem states that all nite division rings are commutative,hence nite elds. The result holds even if we relax associativity and consider alternative rings, by the ArtinZorntheorem.

1.9 See also Quasi-nite eld Field with one element Finite eld arithmetic Trigonometry in Galois elds Finite ring Finite group elementary abelian group Hamming space

1.10 Notes[1] This notation was introduced by E. H. Moore in an address given in 1893 at the International Mathematical Congress held

in Chicago Mullen & Panario 2013, p. 10.

[2] Moore, E. H. (1896), A doubly-innite system of simple groups, in E. H. Moore, et. al., Mathematical Papers Read atthe International Mathematics Congress Held in Connection with the Worlds Columbian Exposition, Macmillan & Co., pp.208242

[3] NIST, Recommended Elliptic Curves for Government Use, page 3

[4] Jacobson 2009, 4.13

[5] This can be veried by looking at the information on the page provided by the browser.


1.11 References Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed.), Dover Publications, ISBN 978-0-486-47189-1 L. Mullen, Garry; Mummert, Carl (2007), Finite Fields and Applications I, Student Mathematical Library(AMS), ISBN 978-0-8218-4418-2

Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6 Lidl, Rudolf; Niederreiter, Harald (1997), Finite Fields (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4

1.12 External links Finite Fields at Wolfram research.

Chapter 2

Glossary of eld theory

Field theory is the branch of mathematics in which elds are studied. This is a glossary of some terms of the subject.(See eld theory (physics) for the unrelated eld theories in physics.)

2.1 Denition of a eldA eld is a commutative ring (F,+,*) in which 01 and every nonzero element has a multiplicative inverse. In a eldwe thus can perform the operations addition, subtraction, multiplication, and division.The non-zero elements of a eld F form an abelian group under multiplication; this group is typically denoted by F;The ring of polynomials in the variable x with coecients in F is denoted by F[x].

2.2 Basic denitionsCharacteristic The characteristic of the eld F is the smallest positive integer n such that n1 = 0; here n1 stands

for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zerocharacteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbershave characteristic 0, while the nite eld Zp has characteristic p.

Subeld A subeld of a eld F is a subset of F which is closed under the eld operation + and * of F and which,with these operations, forms itself a eld.

Prime eld The prime eld of the eld F is the unique smallest subeld of F.

Extension eld If F is a subeld of E then E is an extension eld of F. We then also say that E/F is a eld extension.

Degree of an extension Given an extension E/F, the eld E can be considered as a vector space over the eld F,and the dimension of this vector space is the degree of the extension, denoted by [E : F].

Finite extension A nite extension is a eld extension whose degree is nite.

Algebraic extension If an element of an extension eld E over F is the root of a non-zero polynomial in F[x],then is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension.

Generating set Given a eld extension E/F and a subset S of E, we write F(S) for the smallest subeld of E thatcontains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations+,,*,/ on the elements of F and S. If E = F(S) we say that E is generated by S over F.

Primitive element An element of an extension eld E over a eld F is called a primitive element if E=F(), thesmallest extension eld containing . Such an extension is called a simple extension.

11

12 CHAPTER 2. GLOSSARY OF FIELD THEORY

Splitting eld A eld extension generated by the complete factorisation of a polynomial.

Normal extension A eld extension generated by the complete factorisation of a set of polynomials.

Separable extension An extension generated by roots of separable polynomials.

Perfect eld A eld such that every nite extension is separable. All elds of characteristic zero, and all nite elds,are perfect.

Imperfect degree Let F be a eld of characteristic p>0; then Fp is a subeld. The degree [F:Fp] is called theimperfect degree of F. The eld F is perfect if and only if its imperfect degree is 1. For example, if F is afunction eld of n variables over a nite eld of characteristic p>0, then its imperfect degree is pn.[1]

Algebraically closed eld A eld F is algebraically closed if every polynomial in F[x] has a root in F; equivalently:every polynomial in F[x] is a product of linear factors.

Algebraic closure An algebraic closure of a eld F is an algebraic extension of F which is algebraically closed.Every eld has an algebraic closure, and it is unique up to an isomorphism that xes F.

Transcendental Those elements of an extension eld of F that are not algebraic over F are transcendental over F.

Algebraically independent elements Elements of an extension eld of F are algebraically independent over F ifthey don't satisfy any non-zero polynomial equation with coecients in F.

Transcendence degree The number of algebraically independent transcendental elements in a eld extension. It isused to dene the dimension of an algebraic variety.

2.3 HomomorphismsField homomorphism A eld homomorphism between two elds E and F is a function

f : E F

such that

f(x + y) = f(x) + f(y)

and

f(xy) = f(x) f(y)

for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x1) = f(x)1 for x in E with x 0, andthat f is injective. Fields, together with these homomorphisms, form a category. Two elds E and F are calledisomorphic if there exists a bijective homomorphism

f : E F.

The two elds are then identical for all practical purposes; however, not necessarily in a unique way. See, forexample, complex conjugation.

2.4 Types of eldsFinite eld A eld with nitely many elements.

Ordered eld A eld with a total order compatible with its operations.

2.5. FIELD EXTENSIONS 13

Rational numbers

Real numbers

Complex numbers

Number eld Finite extension of the eld of rational numbers.

Algebraic numbers The eld of algebraic numbers is the smallest algebraically closed extension of the eld ofrational numbers. Their detailed properties are studied in algebraic number theory.

Quadratic eld A degree-two extension of the rational numbers.

Cyclotomic eld An extension of the rational numbers generated by a root of unity.

Totally real eld A number eld generated by a root of a polynomial, having all its roots real numbers.

Formally real eld

Real closed eld

Global eld A number eld or a function eld of one variable over a nite eld.

Local eld A completion of some global eld (w.r.t. a prime of the integer ring).

Complete eld A eld complete w.r.t. to some valuation.

Pseudo algebraically closed eld A eld in which every variety has a rational point.[2]

Henselian eld A eld satisfying Hensel lemma w.r.t. some valuation. A generalization of complete elds.

Hilbertian eld A eld satisfying Hilberts irreducibility theorem: formally, one for which the projective line is notthin in the sense of Serre.[3][4]

Kroneckerian eld A totally real algebraic number eld or a totally imaginary quadratic extension of a totally realeld.[5]

CM-eld or J-eld An algebraic number eld which is a totally imaginary quadratic extension of a totally realeld.[6]

Linked eld A eld over which no biquaternion algebra is a division algebra.[7]

Frobenius eld A pseudo algebraically closed eld whose absolute Galois group has the embedding property.[8]

2.5 Field extensionsLet E / F be a eld extension.

Algebraic extension An extension in which every element of E is algebraic over F.

Simple extension An extension which is generated by a single element, called a primitive element, or generatingelement.[9] The primitive element theorem classies such extensions.[10]


Normal extension An extension that splits a family of polynomials: every root of the minimal polynomial of anelement of E over F is also in E.

Separable extension An algebraic extension in which the minimal polynomial of every element of E over F is aseparable polynomial, that is, has distinct roots.[11]

Galois extension A normal, separable eld extension.

Primary extension An extension E/F such that the algebraic closure of F in E is purely inseparable over F; equiv-alently, E is linearly disjoint from the separable closure of F.[12]

Purely transcendental extension An extension E/F in which every element of E not in F is transcendental overF.[13][14]

Regular extension An extension E/F such that E is separable over F and F is algebraically closed in E.[12]

Simple radical extension A simple extension E/F generated by a single element satisfying n = b for an elementb of F. In characteristic p, we also take an extension by a root of an ArtinSchreier polynomial to be a simpleradical extension.[15]

Radical extension A tower F = F0 < F1 < < Fk = E where each extension Fi/Fi1 is a simple radicalextension.[15]

Self-regular extension An extension E/F such that EFE is an integral domain.[16]

Totally transcendental extension An extension E/F such that F is algebraically closed in F.[14]

Distinguished class A class C of eld extensions with the three properties[17]

1. If E is a C-extension of F and F is a C-extension of K then E is a C-extension of K.2. If E and F are C-extensions of K in a common overeldM, then the compositum EF is a C-extension of

K.3. If E is a C-extension of F and E>K>F then E is a C-extension of K.

2.6 Galois theoryGalois extension A normal, separable eld extension.

Galois group The automorphism group of a Galois extension. When it is a nite extension, this is a nite group oforder equal to the degree of the extension. Galois groups for innite extensions are pronite groups.

Kummer theory The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theoryof quadratic extensions.

ArtinSchreier theory Covers an exceptional case of Kummer theory, in characteristic p.

Normal basis A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.

Tensor product of elds A dierent foundational piece of algebra, including the compositum operation (join ofelds).

2.7. EXTENSIONS OF GALOIS THEORY 15

2.7 Extensions of Galois theoryInverse problem of Galois theory Given a group G, nd an extension of the rational number or other eld with G

as Galois group.

Dierential Galois theory The subject in which symmetry groups of dierential equations are studied along thelines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Liefounded the theory of Lie groups. It has not, probably, reached denitive form.

Grothendiecks Galois theory Avery abstract approach from algebraic geometry, introduced to study the analogueof the fundamental group.

2.8 References[1] Fried & Jarden (2008) p.45

[2] Fried & Jarden (2008) p.214

[3] Serre (1992) p.19

[4] Schinzel (2000) p.298

[5] Schinzel (2000) p.5

[6] Washington, Lawrence C. (1996). Introduction to Cyclotomic elds (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.

[7] Lam (2005) p.342

[8] Fried & Jarden (2008) p.564

[9] Roman (2007) p.46

[10] Lang (2002) p.243

[11] Fried & Jarden (2008) p.28

[12] Fried & Jarden (2008) p.44

[13] Roman (2007) p.102

[14] Isaacs, I. Martin (1994). Algebra: A Graduate Course. Graduate studies in mathematics 100. American MathematicalSociety. p. 389. ISBN 0-8218-4799-6. ISSN 1065-7339.

[15] Roman (2007) p.273

[16] Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. p. 427. ISBN 1-85233-587-4. Zbl1003.00001.

[17] Lang (2002) p.228

Adamson, Iain T. (1982). Introduction to Field Theory (2nd ed.). Cambridge University Press. ISBN 0-521-28658-1.

Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.

Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67.American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

Lang, Serge (1997). Survey ofDiophantine Geometry. Springer-Verlag. ISBN3-540-61223-8. Zbl 0869.11051. Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556


Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics 158. Springer-Verlag. ISBN 0-387-27678-5.

Serre, Jean-Pierre (1989). Lectures on the Mordell-Weil Theorem. Aspects of Mathematics E15. Translatedand edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn.Zbl 0676.14005.

Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics 1. Jones and Bartlett.ISBN 0-86720-210-6. Zbl 0746.12001.

Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics andIts Applications 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.

Chapter 3

Ordered eld

In mathematics, an ordered eld is a eld together with a total ordering of its elements that is compatible with theeld operations. Historically, the axiomatization of an ordered eld was abstracted gradually from the real numbers,by mathematicians including David Hilbert, Otto Hlder and Hans Hahn. In 1926, this grew eventually into theArtinSchreier theory of ordered elds and formally real elds.An ordered eld necessarily has characteristic 0, all natural numbers, i.e. the elements 0, 1, 1 + 1, 1 + 1 + 1, aredistinct. This implies that an ordered eld necessarily contains an innite number of elements: a nite eld cannotbe ordered.Every subeld of an ordered eld is also an ordered eld in the inherited order. Every ordered eld contains anordered subeld that is isomorphic to the rational numbers. Any Dedekind-complete ordered eld is isomorphic tothe real numbers. Squares are necessarily non-negative in an ordered eld. This implies that the complex numberscannot be ordered since the square of the imaginary unit i is 1. Every ordered eld is a formally real eld.

3.1 DenitionsThere are two equivalent denitions of an ordered eld. The denition of total order appeared rst historicallyand is a rst-order axiomatization of the ordering as a binary predicate. Artin and Schreier gave the denition interms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter ishigher-order, viewing positive cones as maximal prepositive cones provides a larger context in which eld orderingsare extremal partial orderings.

3.1.1 Total orderA eld (F, + ,) together with a total order on F is an ordered eld if the order satises the following properties:

if a b then a + c b + c if 0 a and 0 b then 0 ab

The symbol for multiplication will be henceforth omitted.

3.1.2 Positive coneA prepositive cone or preordering of a eld F is a subset P F that has the following properties:[1]

For x and y in P, both x+y and xy are in P. If x is in F, then x2 is in P. The element 1 is not in P.

17

18 CHAPTER 3. ORDERED FIELD

A preordered eld is a eld equipped with a preordering P. Its non-zero elements P form a subgroup of themultiplicative group of F.If in addition, the set F is the union of P and P, we call P a positive cone of F. The non-zero elements of P arecalled the positive elements of F.An ordered eld is a eld F together with a positive cone P.The preorderings on F are precisely the intersections of families of positive cones on F. The positive cones are themaximal preorderings.[1]

3.1.3 Equivalence of the two denitionsLet F be a eld. There is a bijection between the eld orderings of F and the positive cones of F.Given a eld ordering as in Def 1, the elements such that x 0 forms a positive cone of F. Conversely, given apositive cone P of F as in Def 2, one can associate a total ordering P by setting x P y to mean y x P. This totalordering P satises the properties of Def 1.

3.1.4 FanA fan on F is a preordering T with the property that if S is a subgroup of index 2 in F containing T-{0} and notcontaining 1 then S is an ordering (that is, S is closed under addition).[2]

3.2 Properties of ordered elds

0

0 1 x y 0 ^ x < y ) ax < ay

If x < y and y < z, then x < z. (transitivity) If x < y and z > 0, then xz < yz.

3.3. EXAMPLES OF ORDERED FIELDS 19

x < y a+x < a+y

a

The property x < y ) a+ x < a+ y

If x < y and x,y > 0, then 1/y < 1/x

For every a, b, c, d in F:

Either a 0 a or a 0 a. We are allowed to add inequalities": If a b and c d, then a + c b + d We are allowed to multiply inequalities with positive elements": If a b and 0 c, then ac bc.

1 is positive. (Proof: either 1 is positive or 1 is positive. If 1 is not positive, then 1 is positive, so (1)(1)= 1 is positive, which is a contradiction)

An ordered eld has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the eld hadcharacteristic p > 0, then 1 would be the sum of p 1 ones, but 1 is not positive). In particular, nite eldscannot be ordered.

Squares are non-negative. 0 a2 for all a in F. (Follows by a similar argument to 1 > 0)

Every subeld of an ordered eld is also an ordered eld (inheriting the induced ordering). The smallest subeld isisomorphic to the rationals (as for any other eld of characteristic 0), and the order on this rational subeld is thesame as the order of the rationals themselves. If every element of an ordered eld lies between two elements of itsrational subeld, then the eld is said to be Archimedean. Otherwise, such eld is a non-Archimedean ordered eldand contains innitesimals. For example, the real numbers form an Archimedean eld, but hyperreal numbers forma non-Archimedean eld, because it extends real numbers with elements greater than any standard natural number.[3]

An ordered eld K is isomorphic to the real number eld if every non-empty subset of K with an upper bound in Khas a least upper bound in K. This property implies that the eld is Archimedean.

3.2.1 Vector spaces over an ordered eldVector spaces (particularly, n-spaces) over an ordered eld exhibit some special properties and have some specicstructures, namely: orientation, convexity, and positively-denite inner product. See Real coordinate space#Geometricproperties and uses for discussion of those properties of Rn, which can be generalized to vector spaces over otherordered elds.

3.3 Examples of ordered eldsExamples of ordered elds are:

the rational numbers the real algebraic numbers the computable numbers the real numbers

20 CHAPTER 3. ORDERED FIELD

the eld of real rational functions p(x)q(x) , where p(x) and q(x) are polynomials with real coecients, q(x) 6= 0, can be made into an ordered eld where the polynomial p(x) = x is greater than any constant polynomial,by dening that p(x)q(x) > 0 whenever p0q0 > 0 , for p(x) = p0x

n + and q(x) = q0xm + . This orderedeld is not Archimedean.

The eld of formal Laurent series with real coecientsR((x)) , where x is taken to be innitesimal and positive real closed elds superreal numbers hyperreal numbers

The surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered eld. Everyordered eld can be embedded into the surreal numbers.

3.4 Which elds can be ordered?Every ordered eld is a formally real eld, i.e., 0 cannot be written as a sum of nonzero squares.[4][5]

Conversely, every formally real eld can be equipped with a compatible total order, that will turn it into an orderedeld. (This order need not be uniquely determined.) The proof uses Zorns lemma.[6]

Finite elds and more generally elds of nite characteristic cannot be turned into ordered elds, because in charac-teristic p, the element 1 can be written as a sum of (p 1) squares 12. The complex numbers also cannot be turnedinto an ordered eld, as 1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adicnumbers cannot be ordered, since Q2 contains a square root of 7 and Qp (p > 2) contains a square root of 1 p.

3.5 Topology induced by the orderIf F is equipped with the order topology arising from the total order , then the axioms guarantee that the operations+ and are continuous, so that F is a topological eld.

3.6 Harrison topologyThe Harrison topology is a topology on the set of orderings XF of a formally real eld F. Each order can beregarded as a multiplicative group homomorphism from F onto 1. Giving 1 the discrete topology and 1F theproduct topology induces the subspace topology on XF. The Harrison sets H(a) = fP 2 XF : a 2 Pg form asubbasis for the Harrison topology. The product is a Boolean space (compact, Hausdor and totally disconnected),and XF is a closed subset, hence again Boolean.[7][8]

3.7 Superordered eldsA superordered eld is a totally real eld in which the set of sums of squares forms a fan.[9]

3.8 See also Ordered ring Ordered vector space Preorder eld

3.9. NOTES 21

3.9 Notes[1] Lam (2005) p. 289

[2] Lam (1983) p.39

[3] Bair, Jaques; Henry, Valrie. Implicit dierentiation with microscopes (PDF). University of Liege. Retrieved 2013-05-04.

[4] Lam (2005) p. 41

[5] Lam (2005) p. 232

[6] Lam (2005) p. 236

[7] Lam (2005) p. 271

[8] Lam (1983) pp.1-2

[9] Lam (1983) p.45

3.10 References Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathe-matics 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001

Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67.American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.

Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001

Chapter 4

Perfect eld

In algebra, a eld k is said to be perfect if any one of the following equivalent conditions holds:

Every irreducible polynomial over k has distinct roots. Every irreducible polynomial over k is separable. Every nite extension of k is separable. Every algebraic extension of k is separable. Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power. Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism xxp is anautomorphism of k

The separable closure of k is algebraically closed. Every reduced commutative k-algebra A is a separable algebra; i.e.,AkF is reduced for every eld extensionF/k. (see below)

Otherwise, k is called imperfect.In particular, all elds of characteristic zero and all nite elds are perfect.Perfect elds are signicant because Galois theory over these elds becomes simpler, since the general Galois as-sumption of eld extensions being separable is automatically satised over these elds (see third condition above).More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1](This is equivalent to the above condition every element of k is a pth power for integral domains.)

4.1 ExamplesExamples of perfect elds are:

every eld of characteristic zero, e.g. the eld of rational numbers or the eld of complex numbers; every nite eld, e.g. the eld Fp = Z/pZ where p is a prime number; every algebraically closed eld; the union of a set of perfect elds totally ordered by extension; elds algebraic over a perfect eld.

In fact, most elds that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry incharacteristic p>0. Every imperfect eld is necessarily transcendental over its prime subeld (the minimal subeld),because the latter is perfect. An example of an imperfect eld is

22

4.2. FIELD EXTENSION OVER A PERFECT FIELD 23

the eld k(X) of all rational functions in an indeterminate X , where k has characteristic p>0 (because X hasno p-th root in k(X)).

4.2 Field extension over a perfect eldAny nitely generated eld extension over a perfect eld is separably generated.[2]

4.3 Perfect closure and perfectionOne of the equivalent conditions says that, in characteristic p, a eld adjoined with all pr-th roots (r1) is perfect; itis called the perfect closure of k and usually denoted by kp1 .The perfect closure can be used in a test for separability. More precisely, a commutative k-algebra A is separable ifand only if Ak kp1 is reduced.[3]In terms of universal properties, the perfect closure of a ringA of characteristic p is a perfect ringAp of characteristicp together with a ring homomorphism u : A Ap such that for any other perfect ring B of characteristic p with ahomomorphism v : A B there is a unique homomorphism f : Ap B such that v factors through u (i.e. v = fu).The perfect closure always exists; the proof involves adjoining p-th roots of elements of A", similar to the case ofelds.[4]

The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfectclosure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map : R(A) A such that for any perfect ring B of characteristic p equipped with a map : B A, there is a unique map f :B R(A) such that factors through (i.e. = f). The perfection of A may be constructed as follows. Considerthe projective system

! A! A! A!

where the transition maps are the Frobenius endomorphism. The inverse limit of this system is R(A) and consists ofsequences (x0, x1, ... ) of elements of A such that xpi+1 = xi for all i. The map : R(A) A sends (xi) to x0.[5]

4.4 See also p-ring Quasi-nite eld

4.5 Notes[1] Serre 1979, Section II.4

[2] Matsumura, Theorem 26.2

[3] Cohn 2003, Theorem 11.6.10

[4] Bourbaki 2003, Section V.5.1.4, page 111

[5] Brinon & Conrad 2009, section 4.2

4.6 References Bourbaki, Nicolas (2003), Algebra II, Springer, ISBN 978-3-540-00706-7

24 CHAPTER 4. PERFECT FIELD

Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved2010-02-05

Serre, Jean-Pierre (1979), Local elds, Graduate Texts in Mathematics 67 (2 ed.), Springer-Verlag, ISBN978-0-387-90424-5, MR 554237

Cohn, P.M. (2003), Basic Algebra: Groups, Rings and Fields Matsumura, H (2003), Commutative ring theory, Translated from the Japanese by M. Reid. Cambridge Studiesin Advanced Mathematics 8 (2nd ed.)

4.7 External links Hazewinkel, Michiel, ed. (2001), Perfect eld, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 5

Separable extension

In the subeld of algebra named eld theory, a separable extension is an algebraic eld extension E F such thatfor every 2 E , the minimal polynomial of over F is a separable polynomial (i.e., has distinct roots; see below forthe denition in this context).[1] Otherwise, the extension is called inseparable. There are other equivalent denitionsof the notion of a separable algebraic extension, and these are outlined later in the article.The importance of separable extensions lies in the fundamental role they play in Galois theory in nite characteristic.More specically, a nite degree eld extension is Galois if and only if it is both normal and separable.[2] Sincealgebraic extensions of elds of characteristic zero, and of nite elds, are separable, separability is not an obstaclein most applications of Galois theory.[3][4] For instance, every algebraic (in particular, nite degree) extension of theeld of rational numbers is necessarily separable.Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class ofpurely inseparable extensions, also occurs quite naturally. An algebraic extension E F is a purely inseparableextension if and only if for every 2 E nF , the minimal polynomial of over F is not a separable polynomial (i.e.,does not have distinct roots).[5] For a eld F to possess a non-trivial purely inseparable extension, it must necessarilybe an innite eld of prime characteristic (i.e. specically, imperfect), since any algebraic extension of a perfect eldis necessarily separable.[3]

5.1 Informal discussion

An arbitrary polynomial f with coecients in some eld F is said to have distinct roots if and only if it has deg(f)roots in some extension eld E F . For instance, the polynomial g(X)=X2+1 with real coecients has preciselydeg(g)=2 roots in the complex plane; namely the imaginary unit i, and its additive inverse i, and hence does havedistinct roots. On the other hand, the polynomial h(X)=(X2)2 with real coecients does not have distinct roots; only2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(h)=2 roots.To test if a polynomial has distinct roots, it is not necessary to consider explicitly any eld extension nor to compute theroots: a polynomial has distinct roots if and only if the greatest common divisor of the polynomial and its derivativeis a constant. For instance, the polynomial g(X)=X2+1 in the above paragraph, has 2X as derivative, and, over a eldof characteristic dierent of 2, we have g(X) - (1/2 X) 2X = 1, which proves, by Bzouts identity, that the greatestcommon divisor is a constant. On the other hand, over a eld where 2=0, the greatest common divisor is g, and wehave g(X) = (X+1)2 has 1=1 as double root. On the other hand, the polynomial h does not have distinct roots,whichever is the eld of the coecients, and indeed, h(X)=(X2)2, its derivative is 2 (X2) and divides it, and hencedoes have a factor of the form (X )2 for = 2 ).Although an arbitrary polynomial with rational or real coecients may not have distinct roots, it is natural to ask atthis stage whether or not there exists an irreducible polynomial with rational or real coecients that does not havedistinct roots. The polynomial h(X)=(X2)2 does not have distinct roots but it is not irreducible as it has a non-trivialfactor (X2). In fact, it is true that there is no irreducible polynomial with rational or real coecients that does nothave distinct roots; in the language of eld theory, every algebraic extension of Q or R is separable and hence bothof these elds are perfect.

25

26 CHAPTER 5. SEPARABLE EXTENSION

5.2 Separable and inseparable polynomialsA polynomial f in F[X] is a separable polynomial if and only if every irreducible factor of f in F[X] has distinctroots.[6] The separability of a polynomial depends on the eld in which its coecients are considered to lie; forinstance, if g is an inseparable polynomial in F[X], and one considers a splitting eld, E, for g over F, g is necessarilyseparable in E[X] since an arbitrary irreducible factor of g in E[X] is linear and hence has distinct roots.[1] Despitethis, a separable polynomial h in F[X] must necessarily be separable over every extension eld of F.[7]

Let f in F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditionsfor f to be separable; that is, to have distinct roots:

If E F and 2 E , then (X )2 does not divide f in E[X].[8]

There existsK F such that f has deg(f) roots in K.[8]

f and f ' do not have a common root in any extension eld of F.[9]

f ' is not the zero polynomial.[10]

By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero.Since the formal derivative of a positive degree polynomial can be zero only if the eld has prime characteristic,for an irreducible polynomial to not have distinct roots its coecients must lie in a eld of prime characteristic.More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must thecharacteristic of F be a (non-zero) prime number p, but also f(X)=g(Xp) for some irreducible polynomial g in F[X].[11]By repeated application of this property, it follows that in fact, f(X) = g(Xpn) for a non-negative integer n andsome separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).[12]

By the property noted in the above paragraph, if f is an irreducible (non-zero) polynomial with coecients in theeld F of prime characteristic p, and does not have distinct roots, it is possible to write f(X)=g(Xp). Furthermore,if g(X) = P aiXi , and if the Frobenius endomorphism of F is an automorphism, g may be written as g(X) =P

bpiXi , and in particular, f(X) = g(Xp) = P bpiXpi = (P biXi)p ; a contradiction of the irreducibility of f.

Therefore, if F[X] possesses an inseparable irreducible (non-zero) polynomial, then the Frobenius endomorphism ofF cannot be an automorphism (where F is assumed to have prime characteristic p).[13]

If K is a nite eld of prime characteristic p, and if X is an indeterminant, then the eld of rational functions overK, K(X), is necessarily imperfect. Furthermore, the polynomial f(Y)=YpX is inseparable.[1] (To see this, note thatthere is some extension eldE K(X) in which f has a root ; necessarily, p = X in E. Therefore, working overE, f(Y ) = Y p X = Y p p = (Y )p (the nal equality in the sequence follows from freshmans dream),and f does not have distinct roots.) More generally, if F is any eld of (non-zero) prime characteristic for which theFrobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.[14]

A eld F is perfect if and only if all of its algebraic extensions are separable (in fact, all algebraic extensions of Fare separable if and only if all nite degree extensions of F are separable). By the argument outlined in the aboveparagraphs, it follows that F is perfect if and only if F has characteristic zero, or F has (non-zero) prime characteristicp and the Frobenius endomorphism of F is an automorphism.

5.3 Properties

IfE F is an algebraic eld extension, and if ; 2 E are separable over F, then + and are separableover F. In particular, the set of all elements in E separable over F forms a eld.[15]

If E L F is such that E L and L F are separable extensions, then E F is separable.[16]Conversely, if E F is a separable algebraic extension, and if L is any intermediate eld, then E L andL F are separable extensions.[17]

If E F is a nite degree separable extension, then it has a primitive element; i.e., there exists 2 E withE = F [] . This fact is also known as the primitive element theorem or Artins theorem on primitive elements.

5.4. SEPARABLE EXTENSIONS WITHIN ALGEBRAIC EXTENSIONS 27

5.4 Separable extensions within algebraic extensionsSeparable extensions occur quite naturally within arbitrary algebraic eld extensions. More specically, if E Fis an algebraic extension and if S = f 2 Ej is separable over Fg , then S is the unique intermediate eld thatis separable over F and over which E is purely inseparable.[18] If E F is a nite degree extension, the degree [S: F] is referred to as the separable part of the degree of the extension E F (or the separable degree of E/F),and is often denoted by [E : F] or [E : F].[19] The inseparable degree of E/F is the quotient of the degree bythe separable degree. When the characteristic of F is p > 0, it is a power of p.[20] Since the extension E F isseparable if and only if S = E , it follows that for separable extensions, [E : F]=[E : F], and conversely. If E Fis not separable (i.e., inseparable), then [E : F] is necessarily a non-trivial divisor of [E : F], and the quotient isnecessarily a power of the characteristic of F.[19]

On the other hand, an arbitrary algebraic extensionE F may not possess an intermediate extension K that is purelyinseparable over F and over which E is separable (however, such an intermediate extension does exist when E Fis a nite degree normal extension (in this case, K can be the xed eld of the Galois group of E over F)). If suchan intermediate extension does exist, and if [E : F] is nite, then if S is dened as in the previous paragraph, [E :F]=[S : F]=[E : K].[21] One known proof of this result depends on the primitive element theorem, but there doesexist a proof of this result independent of the primitive element theorem (both proofs use the fact that if K F isa purely inseparable extension, and if f in F[X] is a separable irreducible polynomial, then f remains irreducible inK[X][22]). The equality above ([E : F]=[S : F]=[E : K]) may be used to prove that if E U F is such that [E :F] is nite, then [E : F]=[E : U][U : F].[23]

If F is any eld, the separable closure Fsep of F is the eld of all elements in an algebraic closure of F that areseparable over F. This is the maximal Galois extension of F. By denition, F is perfect if and only if its separable andalgebraic closures coincide (in particular, the notion of a separable closure is only interesting for imperfect elds).

5.5 The denition of separable non-algebraic extension eldsAlthough many important applications of the theory of separable extensions stem from the context of algebraic eldextensions, there are important instances in mathematics where it is protable to study (not necessarily algebraic)separable eld extensions.Let F/k be a eld extension and let p be the characteristic exponent of k .[24] For any eld extension L of k, we writeFL = L k F (cf. Tensor product of elds.) Then F is said to be separable over k if the following equivalentconditions are met:

F p and k are linearly disjoint over kp

Fk1/p is reduced. FL is reduced for all eld extensions L of k.

(In other words, F is separable over k if F is a separable k-algebra.)A separating transcendence basis for F/k is an algebraically independent subset T of F such that F/k(T) is a niteseparable extension. An extension E/k is separable if and only if every nitely generated subextension F/k of E/k hasa separating transcendence basis.[25]

Suppose there is some eld extension L of k such that FL is a domain. Then F is separable over k if and only if theeld of fractions of FL is separable over L.An algebraic element of F is said to be separable over k if its minimal polynomial is separable. If F/k is an algebraicextension, then the following are equivalent.

F is separable over k. F consists of elements that are separable over k. Every subextension of F/k is separable. Every nite subextension of F/k is separable.


If F/k is nite extension, then the following are equivalent.

(i) F is separable over k. (ii) F = k(a1; :::; ar) where a1; :::; ar are separable over k. (iii) In (ii), one can take r = 1: (iv) If K is an algebraic closure of k, then there are precisely [F : k] embeddings F into K which x k. (v) If K is any normal extension of k such that F embeds into K in at least one way, then there are precisely

[F : k] embeddings F into K which x k.

In the above, (iii) is known as the primitive element theorem.Fix the algebraic closure k , and denote by ks the set of all elements of k that are separable over k. ks is then separablealgebraic over k and any separable algebraic subextension of k is contained in ks ; it is called the separable closureof k (inside k ). k is then purely inseparable over ks . Put in another way, k is perfect if and only if k = ks .

5.6 Dierential criteriaThe separability can be studied with the aid of derivations and Khler dierentials. Let F be a nitely generated eldextension of a eld k . Then

dimF Derk(F; F ) tr: degk F

where the equality holds if and only if F is separable over k.In particular, if F/k is an algebraic extension, then Derk(F; F ) = 0 if and only if F/k is separable.[26]

Let D1; :::; Dm be a basis of Derk(F; F ) and a1; :::; am 2 F . Then F is separable algebraic over k(a1; :::; am) ifand only if the matrix Di(aj) is invertible. In particular, when m = tr: degk F , fa1; :::; amg above is called theseparating transcendence basis.

5.7 See also Purely inseparable extension Perfect eld Primitive element theorem Normal extension Galois extension Algebraic closure

5.8 Notes[1] Isaacs, p. 281

[2] Isaacs, Theorem 18.13, p. 282


[4] Isaacs, p. 293

[5] Isaacs, p. 298

5.9. REFERENCES 29

[6] Isaacs, p. 280

[7] Isaacs, Lemma 18.10, p. 281

[8] Isaacs, Lemma 18.7, p. 280


[10] Isaacs, Corollary 19.5, p. 296




[14] Isaacs, p. 299

[15] Isaacs, Lemma 19.15, p. 300




[19] Isaacs, p. 302

[20] Lang 2002, Corollary V.6.2


[22] Isaacs, Lemma 19.20, p. 302


[24] The characteristic exponent of k is 1 if k has characteristic zero; otherwise, it is the characteristic of k.

[25] Fried & Jarden (2008) p.38

[26] Fried & Jarden (2008) p.49

5.9 References Borel, A. Linear algebraic groups, 2nd ed. P.M. Cohn (2003). Basic algebra Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.

I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. pp. 5559. ISBN 0-226-42451-0. Zbl 1001.16500.

M. Nagata (1985). Commutative eld theory: new edition, Shokado. (Japanese) Silverman, Joseph (1993). The Arithmetic of Elliptic Curves. Springer. ISBN 0-387-96203-4.

5.10 External links Hazewinkel, Michiel, ed. (2001), separable extension of a eld k, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4


5.11 Text and image sources, contributors, and licenses5.11.1 Text

Finite eld Source: https://en.wikipedia.org/wiki/Finite_field?oldid=670479395Contributors: AxelBoldt, Bryan Derksen, Zundark, Tar-quin, Toby Bartels, PierreAbbat, Chas zzz brown, Michael Hardy, TakuyaMurata, Karada, J-Wiki, Revolver, Charles Matthews, Dcoet-zee, Joshuabowman, Dysprosia, Fibonacci, Fredrik, Ojigiri~enwiki, Bkell, Wikibot, Tobias Bergemann, Giftlite, Gene Ward Smith,Lethe, Fropu, Dratman, Waltpohl, Dries~enwiki, Pmanderson, Elroch, Andreas Kaufmann, Vivacissamamente, Mike Rosoft, Paul Au-gust, Petrus~enwiki, Zaslav, Army1987, Giraedata, Atlant, Emvee~enwiki, Oleg Alexandrov, Ma Baker, Hypercube~enwiki, Isnow,HannsEwald, OmriSegal, Chobot, Bgwhite, Algebraist, YurikBot, JWB, Dmharvey, Lenthe, KSmrq, DYLAN LENNON~enwiki, Hv,SmackBot, Reedy, Zeycus, Gilliam, Chris the speller, Bluebot, Alan smithee, Nbarth, Daqu, BlackFingoln, Dgessner, MvH, JoshuaZ,Noegenesis, Schildt.a, WAREL, VoiceOfOdin, Az1568, Floridi~enwiki, Shernren, Thijs!bot, A3RO, RobHar, Klausness, Urdutext,AntiVandalBot, Lovibond, Magioladitis, Vanish2, JamesBWatson, Albmont, Brusegadi, JoergenB, Cpiral, Maproom, Jacksonwalters,VolkovBot, Safemariner, TXiKiBoT, MichaelShoemaker, SieBot, Thehotelambush, JackSchmidt, Mild Bill Hiccup, Auntof6, 800km3rk,Bender2k14, Sun Creator, SchreiberBike, MagnusPI, Leonid 01, Sahmosavian1, SilvonenBot, MystBot, Addbot, Download, LinkFA-Bot, Legobot, Luckas-bot, Yobot, AnomieBOT, Greggles612, ArthurBot, LilHelpa, Drilnoth, Etoombs, Anne Bauval, Point-set topol-ogist, Sonoluminesence, FrescoBot, Turbano, RedAcer, AaronEmi, Bj norge, Racerx11, Lfrazier11, Quondum, D.Lazard, Wcherowi,Udcep, Hannob, Frietjes, Joel B. Lewis, BG19bot, IkamusumeFan, DNarvaez, MathKnight-at-TAU, Dexbot, Citizentoad, Jamesmath,Cyrapas, K9re11, Spencer m67, Velociraptor 11235813, Rkinser, Chumpih, Teddyktchan, Dyott, GeoreyT2000, Parkerf, Joseph2302,Some1Redirects4You and Anonymous: 116

Glossary of eld theory Source: https://en.wikipedia.org/wiki/Glossary_of_field_theory?oldid=648502002 Contributors: AxelBoldt,Michael Hardy, Wshun, Dcljr, Loren Rosen, CharlesMatthews, Dysprosia, Robbot, Fropu, D6, Rich Farmbrough, Ruud Koot, Marudub-shinki, Marco Streng, Dmharvey, SmackBot, Xyzzyplugh, Cydebot, Nick Number, STBot, CopyToWiktionaryBot, Barylior, Addbot,Bte99, Yobot, Erik9bot, Fred Gandt, CaroleHenson, Deltahedron and Anonymous: 8

Ordered eld Source: https://en.wikipedia.org/wiki/Ordered_field?oldid=667273377Contributors: AxelBoldt, Zundark,Michael Hardy,Charles Matthews, Jitse Niesen, David Shay, SirJective, Aleph4, Tobias Bergemann, Lady Tenar, Giftlite, Gene Ward Smith, Lethe, Li-Daobing, Shreevatsa, Graham87, Salix alba, FlaBot, VKokielov, Grubber, Arthur Rubin, SmackBot, Incnis Mrsi, Melchoir, 10lbs ofpotatoes, Henning Makholm, Mets501, Zero sharp, Andorin, Dr.enh, Oerjan, RobHar, Dugwiki, Asmeurer, Magioladitis, Albmont,David Eppstein, Ueno Rad, Broadbot, AlleborgoBot, JackSchmidt, Malatinszky, DesolateReality, Randomblue, Niceguyedc, WikHead,Tassedethe, Legobot, Luckas-bot, Yobot, Ptbotgourou, Legobot II, Citation bot, Omnipaedista, Tkuvho, WikitanvirBot, ZroBot, Jcim-pric, Deltahedron, Spectral sequence, Stephan Kulla, Mark viking, DG-on-WP and Anonymous: 23

Perfect eld Source: https://en.wikipedia.org/wiki/Perfect_field?oldid=587177721 Contributors: AxelBoldt, Zundark, TakuyaMurata,Charles Matthews, EmilJ, CBM, Myasuda, Konradek, Headbomb, RobHar, Albmont, Policron, Addbot, Roentgenium111, Legobot,Yobot, Ptbotgourou, Darij, Brained, Quondum, Helpful Pixie Bot, CitationCleanerBot, Danneks and Anonymous: 7

Separable extension Source: https://en.wikipedia.org/wiki/Separable_extension?oldid=659808099 Contributors: AxelBoldt, Zundark,Edward, TakuyaMurata, Charles Matthews, Dysprosia, Vivacissamamente, Shotwell, Mazi, Bender235, Vipul, EmilJ, Keenan Pepper,OlegAlexandrov, R.e.b., YurikBot, SmackBot, Eskimbot, Mets501,WLior, Tac-Tics, Dragonare82, Thijs!bot, RobHar, Jakob.scholbach,Etale, Allispaul, LokiClock,Wedhorn, Ideal gas equation, Niceguyedc, Alexbot, Bender2k14, Addbot, Luckas-bot, FredrikMeyer, AnomieBOT,Citation bot, LilHelpa, GrouchoBot, Point-set topologist, Hkhk59333, Citation bot 1, John of Reading, Codygunton, D.Lazard, Ben-delacBOT, Deltahedron, Mark viking and Anonymous: 23

5.11.2 Images File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: The

Tango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the le, specically: Andreas Nilsson, and Jakub Steiner (althoughminimally).

File:Invariance_of_less-than-relation_by_multiplication_with_positive_number.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/03/Invariance_of_less-than-relation_by_multiplication_with_positive_number.svg License: CC BY 3.0 Contributors: Ownwork Original artist: Stephan Kulla (User:Stephan Kulla)

File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

File:Translation_invariance_of_less-than-relation.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1b/Translation_invariance_of_less-than-relation.svg License: CC BY 3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)

5.11.3 Content license Creative Commons Attribution-Share Alike 3.0

Finite fieldDefinitions, first examples, and basic properties Existence and uniquenessExplicit construction of finite fields Non-prime fieldsField with four elementsGF(p2) for an odd prime pGF(8) and GF(27)GF(16)

Multiplicative structure Discrete logarithm Roots of unityExample

Frobenius automorphism and Galois theory Polynomial factorizationIrreducible polynomials of a given degreeNumber of monic irreducible polynomials of a given degree over a finite field

Applications Extensions Algebraic closureWedderburns little theorem

See also NotesReferences External links

Glossary of field theoryDefinition of a fieldBasic definitionsHomomorphismsTypes of fieldsField extensionsGalois theoryExtensions of Galois theoryReferences

Ordered fieldDefinitionsTotal orderPositive coneEquivalence of the two definitions Fan

Properties of ordered fieldsVector spaces over an ordered field

Examples of ordered fields Which fields can be ordered?Topology induced by the order Harrison topologySuperordered fieldsSee alsoNotesReferences

Perfect fieldExamplesField extension over a perfect field Perfect closure and perfectionSee also NotesReferences External links

Separable extensionInformal discussionSeparable and inseparable polynomialsPropertiesSeparable extensions within algebraic extensionsThe definition of separable non-algebraic extension fields Differential criteria See alsoNotesReferencesExternal linksText and image sources, contributors, and licensesTextImagesContent license

ordered field

Documents

galois eld

agiven eld

nite eld of order q

eld of order pk

glossary of eld theory

q elements

given order

prime elds