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What is reciprocity ?

On the evolution of class field theory in the20th century.

by Peter Roquette

August 9, 2013

2

This is a preliminary manuscript which grew out of a talk givenon April 15, 2010 at Ovidius University, Constantza at the In-ternational Conference “Fundamental Structures of Algebra”. Itwill be continued in the course of time.

Contents

1 Preface 5

2 The Gottingen prize 7

3 The reciprocity problem 11

3.1 Quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Cubic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Biquadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Artin and the consequences 21

4.1 Artin’s reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Furtwangler’s wonderful proof . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Hilbert’s product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 On infinite primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Hasse’s norm symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.1 Grunwald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 Chevalley’s ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4.1 Chevalley’s encyclopedia article . . . . . . . . . . . . . . . . . . . . . 46

4.4.2 Chevalley’s final paper . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.3 On the name “idele” . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 The Highlights 53

5.1 The Local-Global Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3

4 CONTENTS

5.2 Simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Local theory 61

6.1 Local class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Appendix: About Furtwangler 63

References 68

Chapter 1

Preface

Mathematics is a cumulative science. Once a mathematical theorem has beenproved to be valid then it remains valid forever; it is added to the stock ofmathematical discoveries which has piled up through the centuries and it canbe used to proceed still further in our pursuit of knowledge.

But the mere proof of the validity of a theorem is in general not satisfac-tory to mathematicians. We also want to know “why” the theorem is true,we strive to gain a better understanding of the situation than was possiblefor previous generations. Sometimes a result seems to be better understoodif it is generalized, or if it is looked at from a different point of view, or ifit is embedded into a general theory which opens analogies to other fieldsof mathematics. Also, in order to make further progress possible it is oftenconvenient and sometimes necessary to develop a framework, conceptual andnotational, in which the known result becomes trivial and self-evident at leastfrom a formalistic point of view.

So when we look at the history of mathematics we observe a continuouschange in the attitude of mathematicians towards the results of their science.It may happen that the next generation will not be able to easily recognizethe highlights of the past simply because the mathematical environment hasbeen extended and the terminology has changed accordingly. There are manyexamples for such situation.1 In this book we shall discuss the example ofthe reciprocity law of number theory, as far as it is possible for us in these

1An impressive discussion of the development of such situation can be found in Lakatos’book [Lak76], with the example of Euler’s polyhedron formula.

5

6 CHAPTER 1. PREFACE

limited pages.

The development of reciprocity and class field theory in the first half ofthe 20th century has been a brilliant feat of several generations of mathe-maticians. The roots of the problem extend far back into the 18th and 19thcentury but in the year 1898 Hilbert definitely formulated the general prob-lem. He repeated it 1900 in his famous Paris lecture where he included it intohis 23 most important problems for the 20th century. This started an inten-sive activity of few but highly motivated mathematicians of the first rank,searching for an adequate viewpoint to understand the arithmetic structureof reciprocity, thereby opening a wide and rich field of research. The namesinclude Furtwangler, Takagi, Artin, Hasse, Chevalley and others.

It is fascinating to observe this activity, the growth of class field theory inthat period, the line of ideas which were handed down from one generation tothe next, until the theory had reached the form which is familiar to us. This isa fine example of mathematical research as a collective effort, not planned byany research authority or agency but making headway just by the dedicationof the people and their interest in the subject, combined with distributing theresults freely and openly to colleagues and to the mathematical community.

In the first part of this book (until chapter 4) we will give a brief overviewof the main lines of development, starting from Furtwangler’s work in 1901until Chevalley’s in 1940. This part grew out of a talk given on April 15, 2010at Ovidius University, Constantza at the International Conference “Funda-mental Structures of Algebra”. This part is addressed to a general mathe-matical audience and no details of proof are given.

The second part (starting from chapter 5) presents in some more detailthe highlights in this development, including some details of proof. In thispart we suppose that the reader will be somewhat familiar with at least thebasic concepts and results of algebraic number theory.

Chapter 2

The Gottingen prize

Our story begins in the year 1898. In that year there appeared a note inthe Mathematische Annalen with an announcement of the Royal Society forScience in Gottingen1 of a prize problem about reciprocity [Pri98]. Theprize was endowed with one thousand German Mark – at that time a sizableamount.2 The deadline for presenting the paper to the Society was set asFebruary 1, 1901. The text of the announcement started as follows:

Es soll fur einen beliebigen Zahlkorper das Reziprozitatsgesetz der`-ten Potenzreste entwickelt werden, wenn ` eine ungerade Primzahlbedeutet . . .

For an arbitrary number field the reciprocity law for the `-th powerresidues is to be developed, where ` is an odd prime number . . . 3

There followed some explanations about the problem and what was ex-pected. The relevant chapters of Hilbert’s Zahlbericht [Hil97] were cited,in which the problem was solved for the `-th cyclotomic field for regularprimes `. It seems that Hilbert himself, being a member, had induced the

1Konigliche Gesellschaft der Wissenschaften zu Gottingen.2But this sum was much less than Paul Fiedrich Wolfskehl had donated for the proof of

Fermat’s “Last Theorem”. The Wolfskehl prize was announced by the Gottingen Societyin 1908 and amounted to 100000 Mark. The time limit for the Wolfskehl prize was 100years.

3In this context the letter ` is usually used to denote a prime number.

7

8 CHAPTER 2. THE GOTTINGEN PRIZE

Society to donate this prize. Hilbert had already treated the quadratic casein two papers [Hil99b], [Hil98]. This explains why the prize problem was putforward for odd prime numbers ` only.

Hilbert considered the problem to be of such an order of magnitude thathe repeated it, even before expiration of the above mentioned deadline, inhis famous address at the ICM in Paris 1900. He included the reciprocityproblem among his 23 mathematical problems which he presented as beingthe most important ones for Mathematics in the coming 20-th century, in hisopinion. The reciprocity problem became the 9-th in Hilbert’s enumeration.There we read:

Fur einen beliebigen Zahlkorper soll das Reciprocitatsgesetz der `-tenPotenzreste bewiesen werden, wenn ` eine ungerade Primzahl be-deutet und ferner, wenn ` eine Potenz von 2 oder eine Potenz einerungeraden Primzahl ist.

For any field of numbers the law of reciprocity is to be proved forthe residues of the `-th power, when ` denotes an odd prime, andfurther when ` is a power of 2 or a power of an odd prime.4

We see that Hilbert had upgraded his original problem since now he men-tioned the reciprocity law not only for primes ` but also for prime powers.

In January 1901, less than one year after Hilbert’s Paris address and intime with the deadline for the prize, the secretary of the Gottingen Societyreceived a manuscript with a solution of the prize problem – albeit under therestrictive assumption that the number field in question has class numbernot divisible by the prime number `. Nevertheless, despite this restrictionthe Society decided to award the full prize to the author, after Hilbert hadbeen consulted and had given his recommendation. His review is publishedin [Pri01]. There we read:

. . . Die eingereichte Arbeit erscheint als eine in hohem Maße anzu-erkennende Leistung ihres Verfassers und zeugt von einem schonenzahlentheoretischen Talent.

. . . The submitted paper appears as an accomplishment which isto be highly appreciated. It shows a fine number theoretic talentof the author.

4English translation from [Hil02].

9

The name of the author was Philipp Furtwangler.

The said restrictive assumption about the class number seemed to besomewhat natural since Hilbert himself, in the case of quadratic reciprocity,also had first to assume that the class number of the field is odd. In a secondattempt only he had been able to show how to remove that restriction.

The author of the prize paper too did not stop his work and started asecond atempt; and indeed he was able to remove the restriction in a series ofpapers, culminating with a full solution in the year 1912 [Fur12]. To be sure,this was the solution of the prize problem of 1898 concerning the `-powerreciprocity law just for a prime number `, and not yet for arbitrary primepowers `n.

Before starting our story about the reciprocity problem and what becameof it under the hands of Furtwangler and of Takagi, Artin, Hasse and others,let us insert some explanations of elementary nature.

What actually is the reciprocity problem in the sense of Hilbert?

10 CHAPTER 2. THE GOTTINGEN PRIZE

Chapter 3

The reciprocity problem

3.1 Quadratic reciprocity

Let p be an odd prime number and a an integer not divisible by p. If theresidue class of a modulo p is a square in the residue field Z/p then a is called

a quadratic residue modulo p and we write(ap

)= 1. Otherwise, we write(

ap

)= −1. This is the so-called Legendre symbol. We have Euler’s formula(

a

p

)≡ a

p−12 mod p (3.1)

as a consequence of the fact that the multiplicative group of the residue fieldZ/p is cyclic of order p− 1. This formula can also be taken as the definition

of the Legendre symbol(ap

)∈ {−1, 1}.

The quadratic reciprocity law compares the Legendre symbol(qp

)with

its “reciproque”(pq

). It reads(

p

q

)=

(q

p

)if p or q ≡ 1 mod 4 . 1 (3.2)

1This means that p ≡ 1 mod 4 or q ≡ 1 mod 4. We shall use this simplified expressionalso in similar situations below.

11

12 CHAPTER 3. THE RECIPROCITY PROBLEM

Of course it is assumed that both Legendre symbols are defined, i.e., thatp, q are different odd prime numbers.

Thus if the stated congruence condition is satisfied then the Legendresymbol is self-reciprocal; this explains the terminology “reciprocity law”.

For the history of the quadratic and higher reciprocity laws in the 19thcentury we refer to the excellent presentation in the book of Lemmermeyer[Lem00]. There we read that Gauss had given the first complete proof ofthe quadratic reciprocity law; an earlier proof by Legendre was not quitecomplete since he used auxiliary prime numbers whose existence he couldnot yet prove. Gauss’ proof is contained in his Disquisitiones arithmeticae,published in 1801 when Gauss was 24 years old, but he had discovered italready six years earlier. Gauss called it the “‘theorema aureum”, the “goldentheorem”, and he gave at least 6 proofs during his lifetime. It is said thattoday there are more than 150 proofs 2 – but we do not want to open adiscussion about when two proofs are to be considered as different. In anycase, the fact that so many mathematicians have tried to provide a new proofshows that the reciprocity law has been recognized to be very important,and hence people wished to look at it from various viewpoints in order tounderstand its role in number theory and in mathematics at large.

Usually, when we teach a beginner’s course on number theory then thequadratic reciprocity law is included (hopefully) and it is stated in a some-what more general form: (

p

q

)=

(q

p

)(−1)

p−12

q−12 (3.3)

without any congruence condition as in (3.2). In other words: If the congru-ence condition in (3.2) is not satisfied then the Legendre symbol is anti-self-reciprocal.

It is easily seen that (−1

p

)= (−1)

p−12 . (3.4)

This is usually called the “first complementary law”. There is also a second

2This number appears in the literature and is cited in [Lem00].

3.1. QUADRATIC RECIPROCITY 13

complementary law:(2

p

)= (−1)

p2−18 =

{1 if p ≡ ±1 mod 8

−1 if p ≡ ±5 mod 8(3.5)

From Euler’s formula (3.1) is is evident that the Legendre symbol is mul-tiplicative with respect to its numerator:(

a1a2

p

)=

(a1

p

)(a2

p

)provided these symbols are defined, i.e., both a1 and a2 are not divisible by p.Jacobi has extended the Legendre symbol such that it is multiplicative alsoin its denominator. If b is any odd number with the prime decomposition

b = ±∏ν

prνν (3.6)

then the Jacobi symbol is defined by multiplicativity:(ab

)=∏ν

( apν

)rν. (3.7)

This Jacobi symbol is multiplicative in each of the two variables a amd b. Itis defined for arbitrary non-zero rational numbers a and b, provided a and bare relatively prime to each other3 and b is odd4. The reciprocity law for theJacobi symbol reads:(a

b

)=( ba

)if a or b ≡ 1 mod 4 and a or b > 0 , (3.8)

provided both symbols on the left and on the right hand side are defined.

For computations it is often convenient to use the Jacobi symbol insteadof the Legendre symbol. But the reciprocity law for the Jacobi symbol isessentially the same as for the Legendre symbol. It is straightforward toderive (3.8) from (3.2) by means of the definition (3.7); this can be given asa problem to students in a beginners course.

3I.e., the set of primes occurring in a is disjoint to the set of primes occurring in b.4I.e., the prime 2 does not occurr in b.


The “general” reciprocity law for the Jacobi symbol analoguous to (3.3)reads as follows: (a

b

)( ba

)−1

= (−1)a−12

b−12 (−1)

sgn(a)−12

sgn(b)−12 (3.9)

provided a and b are odd and relatively prime to each other. The signfunction sgn(a) is 1 or −1 according to whether a > 0 or a < 0. The signdoes not appear in (3.3) since there p and q are supposed to be positive primenumbers.

3.2 Cubic reciprocity

The quadratic reciprocity law had its origin in the question: Which numbersare squares modulo p ? Similarly one can ask: Which numbers are cubesmodulo p ?

Experience has shown that to discuss this question it is advisable toextend the rational number field by adjoining the cubic roots of unity. Thegroup of cubic roots of unity is of order 3 and contains the elements 1, %, %2

where % = −1+√−3

2. So we shall work in the quadratic field Q(%).

The ring of integers in Q(%) is Z[%]. This is a principal ideal domain. Weuse the notation a ∼= b if the elements a, b 6= 0 differ by a unit factor only.5

The prime elements π in Z[%] can be described as follows.

Let p be the characteristic of the residue field Z[%]/π. Let us write N (π)for the norm of π.

p ≡ 1 mod 3 : Then N (π) = p and p splits, i.e., p ∼= ππ is the product oftwo primes of Z[%] with π 6∼= π.

p ≡ 2 mod 3 : Then N (π) = p2 and p is inert, i.e., p ∼= π remains prime inZ[%].

p = 3 : Then π ∼=√−3 and 3 is ramified, i.e., 3 ∼= π2. We have N (π) = 3.

N (π) equals the number of elements in the residue field Z[%]/π, and inthe first two cases we have N (π) ≡ 1 mod 3. Hence the following definitionof the cubic residue symbol makes sense.

5The units in Z[%] are ±1,±%,±%2.

3.2. CUBIC RECIPROCITY 15

Let π be a prime in Z[%] relatively prime to 3, and a ∈ Z[%] not divisibleby π. Then

(aπ

)3

is defined to be unique cubic root of unity which satisfiesthe congruence (a

π

)3≡ a

N (π)−13 mod π. (3.10)

This is the analogue of Euler’s formula (3.1) for the quadratic residue symbol.a is a cube modulo π if and only if

(aπ

)3

= 1.

As in the quadratic case we have multiplicativity:(a1a2

π

)3

=(a1

π

)3

(a2

π

)3.

Jacobi’s extension leads to a symbol(ab

)3

which is bi-multiplicative: If b isrelatively prime to 3 and if

b ∼=∏ν

πrνν

is its prime decomposition then we put(ab

)3

=∏ν

( aπν

)rν3

Here, a and b can be arbitrary non-zero elements of the field Q(%), with thespecification that b is relatively prime to 3 and a, b are relatively prime toeach other.

The cubic reciprocity law for the Jacobi symbol reads as follows:(ab

)3

=( ba

)3

if a or b ≡ 1 mod 3√−3 , (3.11)

provided both cubic residue symbols are defined, i.e., a, b are relatively primeto each other and to 3. As in the case of the quadratic reciprocity law (3.8)there appears a congruence condition on the right hand side. But this timethere is no positivity condition; note that the quadratic field field Q(%) isimaginary and hence cannot be ordered.

The cubic reciprocity law had first been proved by Eisenstein in 1844.6

There is also a “general” law expressing explicitly the multiplicative differ-

ence(ab

)3

(ba

)−1

3if the congruence condition in (3.11) is not satisfied, like (3.3)

6We read in [Lem00] that Jacobi in his Konigsberg lectures had in 1837 already givena proof. See [Jac07].


in the quadratic case. And there is also a “second complementary law” ex-

pressing(√−3π

)3

by an explicit formula, analoguous to (3.5) in the quadratic

case. But we do not wish to go into these details here.

3.3 Biquadratic reciprocity

In order to study biquadratic power residues we extend Q by adjoining thefourth roots of unity, i.e., we work in the Gaussian field Q(i) with i =

√−1.

The ring of integers in this field is Z[i]. This is a principal ideal domain.Again we write a ∼= b if a and b differ by a unit factor only.7 The primeelements π in Z[i] can be described as follows.

Let p be the characteristic of the residue field Z[i]/π.

p ≡ 1 mod 4 : Then N (π) = p and p splits, i.e., p ∼= ππ is the product oftwo primes of Z[i] with π 6∼= π.

p ≡ 3 mod 4 : Then N (π) = p2 and p is inert, i.e., p ∼= π remains prime inZ[i].

p = 2 : Then π ∼= 1 + i and 2 is ramified, i.e., 2 ∼= π2. We have N (π) = 2.

In the first two cases we have N (π) ≡ 1 mod 4. Now we proceed similarlyas in the quadratic and cubic case:

Let π be a prime in Z[i] relatively prime to 2 and a ∈ Z[i] not divisibleby π. Then

(aπ

)4

is defined to be the unique 4-th root of unity which satisfiesthe congruence (a

π

)4≡ a

N (π)−14 mod π. (3.12)

This is the analogue of Euler’s formula. a is a 4-th power modulo π if andonly if

(aπ

)4

= 1.

Again we have (a1a2

π

)4

=(a1

π

)4

(a2

π

)4.

7The units in Z[i] are ±1,±i.

3.4. GENERALIZATION 17

Jacobi’s extension leads to a symbol(ab

)4

which is bi-multiplicative: If b isrelatively prime to 2 and if

b ∼=∏ν

πrνν

is its prime decomposition then we put(ab

)4

=∏ν

( aπν

)rν4.

Here, a and b can be arbitrary elements of the field Q(i), with the specificationthat b is relatively prime to 2 and a, b are relatively prime to each other. Thebiquadratic reciprocity law for the Jacobi symbol reads as follows:(a

b

)4

=( ba

)4

if a or b ≡ 1 mod (1 + i)4 , (3.13)

provided both biquadratic residue symbols are defined. As in the case of thequadratic and the cubic reciprocity laws there appears a congruence conditionon the right hand side.

3.4 Generalization

After quadratic, cubic and biquadratic reciprocity it would be possible tocontinue with quintic reciprocity and more. During much of the 19-th cen-tury, people have tried to obtain reciprocity laws in various situations. Fromtoday’s viewpoint these are special cases of the general reciprocity law for anarbitrary exponent m, which we are now going to explain.

At the same time we will discuss the generalization in another direction:Namely we work over an arbitrary number field K containing the m-th rootsof unity, not necessarily the cyclotomic field Q( m

√1) itself. This is an impor-

tant step; it had first been taken by Hilbert in the case m = 2 which alreadyturned out to be highly nontrivial [Hil99b], [Hil98]. And then in his 9-thproblem again Hilbert envisaged explicitly the development of reciprocity“for arbitrary number fields”. Of course, in this context he understood “ar-bitrary number field” in the sense of “arbitrary number field containing therelevant roots of unity”. At that time the notion of reciprocity was connectedwith the Legendre power residue symbol and its generalization by Jacobi; this


would not make sense if the proper roots of unity were not contained in thefield.8 Hasse writes about this in his report [Has32c]:

Hilbert’s Schritt zu allgemeinen algebraischen Grundkorpern bedeutetals neue Zielsetzung das Studium der algebraischen Zahlkorper umihrer selbst willen . . . Bisher hatte man sich doch durchweg auf dieBetrachtung der gerade erforderlichen Zahlkorper beschrankt, imFalle des Reziprozitatsgesetzes also des m-ten Kreiskorpers . . .

Hilbert’s move towards arbitrary algebraic base fields signifies anew objective, namely the investigation of algebraic number fieldsfor their own sake . . . Before, one had limited oneself to considerjust the number fields necessary for the particular problem, hencein the case of the reciprocity law the m-th cyclotomic field . . .

So let m be a natural number > 1, and K an arbitrary algebraic numberfield containing the m-th roots of unity. In general the ring R of integersin K will not be a principal ideal domain. Hence we have to work withprime ideals P of R instead of prime elements. We denote by NP theabsolute norm of P , i.e., the number of elements in the residue field R/P .If P does not divide m then NP ≡ 1 mod m. If this is the case then for anya 6≡ 0 mod P the m-th power residue symbol is defined by the Euler formula( a

P

)≡ a

NP−1m mod P . (3.14)

More precisely:(aP

)should be the unique m-th root of unity in K which

satisfies the above congruence. This is the m-th Legendre symbol in thefield K. It is defined whenever P - m and P - a.

The relation(aP

)= 1 signifies that a is an m-th power in the residue field

modulo P .

Actually we should have written(aP

)m

instead of(aP

), in order to indicate

that we consider m-th powers, similarly as we had done above for cubicpowers (m = 3) and biquadratic powers (m = 4). Moreover, we should have

8Sometimes in the literature the word “arbitrary” in Hilbert’s 9-th problem is inter-preted as if Hilbert had in mind a reciprocity law without the relevant roots of unity inthe field. But we have found no evidence for this imterpretation. The step towards fieldswithout roots of unity was taken by Takagi and Artin; see section 4.1.

3.4. GENERALIZATION 19

indicated the base field K and thus should have written(aP

)m,K

. But in

order to simplify the notation we omit the subscripts m and K whenever itdoes not lead to misunderstanding.

Again we have for the m-th Legendre symbol:(a1a2

P

)=(a1

P

)(a2

P

).

provided a1 and a2 are both relatively prime to P . Again we use the Jacobiextension of this symbol, but this time not only for elements b but moregenerally for divisors B. A divisor of K is a formal product of prime divisorsof the form

B =∏ν

P rP (3.15)

with integer exponents rP of which only finitely many are nonzero. Supposethat B is relatively prime to a and to m, i.e., that no prime ideal P occurringin B does occur in a or in m. Then we define( a

B

)=∏P

( aP

)rP.

This is the m-th Jacobi symbol in the field K. If B = (b) is the principaldivisor of an element b then we simply write

(ab

). It is defined for 0 6= a, b ∈ K

whenever b is relatively prime tom, and a, b are relatively prime to each other.

There arises the question whether there exists a reciprocity law for thisgeneral Jacobi symbol. Under which condition is

(ab

)=(ba

)?

Today we know the answer. It is the outcome of the study of quite anumber of special cases during the 19-th century, and it has been conjecturedby Hilbert who had proved it for m = 2.

Let 0 6= a ∈ K be relatively prime to m. We say that a is m-primary9

in K if no prime divisor of m is ramified in the cyclic extension K( m√a) ;

moreover a should be totally positive: a� 0.10 If it is clear from the context

9This name is traditional in this context. In other areas of mathematics the terminology“primary” has different meanings.

10This means that a should be positive with respect to every ordering of K. Thiscondition is vacuous if m > 2 because then there are no orderings since K contains them-th roots of unity. In the case K = Q there is a unique ordering of K; accordingly thecondition a > 0 appears in (3.8).


which field K and which exponent m is under consideration then we maybriefly say that “a is primary”. The m-th power reciprocity law for thegeneralized Jacobi symbol now reads as follows:

(ab

)=( ba

)if a or b is m-primary in K. (3.16)

Of course it is assumed that the symbols on both sides are defined, i.e., a, bare relatively prime to m and to each other.

In the case m = 2 with K = Q , the condition to be 2-primary is explicitlygiven by the condition we have already met in (3.8) (up to a square). Inthe case m = 3 with K = Q(

√%) then a is 3-primary if and only if the

congruence condition in (3.11) is satisfied (up to a cube). In the case m = 4our condition in (3.13) implies that a is 4-primary in Q(i) but the precisecondition for being 4-primary is somewhat involved and cannot be statedsolely by such congruence condition as in the case m = 3.

Chapter 4

Artin and the consequences

4.1 Artin’s reciprocity law

The reciprocity law (3.16) was conjectured by Hilbert for all algebraic numberfields K containing the m-th roots of unity. In the quadratic case (m = 2) hehad proved it in 1898/99. Furtwangler proved it for an odd prime (m = `)in 1909-12.1 Takagi proved this again in 1922, in the framework of class fieldtheory which he had earlier developed from a new viewpoint. For arbitrary mthis reciprocity law was proved in 1927, but only after Artin had obtainedthe theorem which he called the “general reciprocity law” and is now knownas “Artin’s reciprocity law” [Art27].

Actually, Artin’s reciprocity law belongs to class field theory. For anyabelian extension L|K , it establishes a canonical isomorphism between itsGalois group with a certain divisor class group in the field K. As such it doesnot directly refer to reciprocity in the classical sense which we have explainedin chapter 3. But Artin had clearly seen that his isomorphism is the key tosolve Hilbert’s reciprocity problem, and so he considered his theorem as beingthe “true” formulation of the reciprocity law. This explains why he called hisresult the “general reciprocity law”. But he admitted in his paper [Art23] 2

that on first sight this name for his theorem might sound somewhat strange

1Hilbert’s proof for m = 2 was somewhat sketchy in the case when the class number ofK is even. Therefore Furtwangler provided a detailed proof also for m = 2 [Fur13].

2This paper appeared in 1923 already. There, Artin had stated already his reciprocitylaw but without yet being able to give a general proof.

21

22 CHAPTER 4. ARTIN AND THE CONSEQUENCES

to his contemporaries, who were used to connect the name “reciprocity” withthe classical problem (3.16).

The appearance of Artin’s proof in 1927 marked a change of paradigm indealing with reciprocity. Mathematicians began to regard the classical reci-procity problem as only one of various applications of class field theory whichin the hands of Takagi had been rebuilt as the theory of abelian extensionsof number fields, and which Artin had topped with his reciprocity law. Classfield theory now had become the main object of study, and it inherited thename “reciprocity”.

In order to avoid confusion we shall denote Artin’s theorem explicitlyas “Artin’s reciprocity law” whereas we use “classical reciprocity” or just“reciprocity” when we refer to the classical problem formulated by Hilbert.

The main aim of Artin’s paper [Art27] is the proof of Artin’s reciprocitylaw, while the proof of the classical reciprocity formula (3.16) is found at theend of the paper only, as kind of corollary. We know from the Artin-Hassecorrespondence that Hasse had shown him how to obtain (3.16) from Artin’sreciprocity law, by means of a simple argument due to Furtwangler. Artincites Hasse and Furtwangler in his paper. See section 4.1.1 below.

But what precisely does Artin’s reciprocity law say?

Artin worked in the framework of Takagi’s class field theory. Takagi hadshown that every abelian extension L of a number field K can be character-ized by a certain divisor group H which is obtained as follows.

Let S be a finite set of primes of K containing all primes Q which areramified in L, and also the infinite primes.3 Consider the group DS of thosedivisors of K which are composed of primes P /∈ S. These primes P areunramified in L. Takagi’s group H is contained in DS. It is the groupgenerated by:

1. those divisors in DS which are norms from divisors of L, and

2. those principal divisors (a) ∈ DS which can be generated by elementsa ∈ K× which are locally norms from L×Q for every Q ∈ S.

Class field theory contains the following theorem:

3For the infinite primes see page 33 ff.

4.1. ARTIN’S RECIPROCITY LAW 23

For each prime P /∈ S (which by construction is unramified in L)its type of decomposition in L is already determined by its classin the associated factor group DS/H.

Specifically, if fP is the order of the class of P modulo H then Pdecomposes in L into primes of relativ degree fP , and the numberof prime factors of P in L is n/fP where n = [L : K] is the fielddegree. In particular the primes P of order fP = 1, i.e., P ∈ H,are split completely in L.

This characterizes the field L as an extension of K.

The appearance of the word “class” in the above theorem is responsiblefor the terminology that L is a “class field” over K. Accordingly the abovetheorem can be rephrased as:

Takagi’s theorem. Every finite abelian extension L of K is aclass field over K .

Of course, the definition of H depends on the choice of the finite set Swhich may be arbitrary with the only condition that it contains the ramifiedand the infinite primes. But for any such choice, the class field theoremholds, and one proves that the factor group DS/H is isomorphic to the Galoisgroup G of L|K. Artin’s reciprocity law tops this statement by providing a

canonical isomorphism DS/H≈→ G, as follows.

Artin’s reciprocity law: For each P /∈ S let σP ∈ G denoteits Frobenius automorphism. Extend the map P 7→ σP by mul-tiplicativity to a homomorphism B 7→ σB for divisors B ∈ DS.This map σ : DS → G, which is called the Artin map, yields anisomorphism from the factor group DS/H to the Galois group Gof L|K.

In other words: The kernel of the Artin map is H and its image is G. Inparticular it follows that σB depends only on the class of B modulo H. Infact, the proof of this very statement is the crucial part of Artin’s proof ofhis reciprocity law in [Art27].


Takagi wrote in his review of Artin’s paper:4

Jedenfalls erscheint mir dieser Artikel Artins als eines der schon-sten der in letzter Zeit gewonnenen Ergebnisse der algebraischenZahlentheorie.

In any case I regard this article as one of the most beautiful resultsof algebraic number theory which have been obtained in recenttimes.

As we have already said, on first sight Artin’s reciprocity law does notseem to have any bearing to the classic reciprocity law (3.16). But by lookingcloser into the situation one finds indeed a connection between Artin’s σBand the Jacobi symbol

(aB

). Let us explain:

Let m be a given positive integer and assume that K contains the m-throots of unity. For a ∈ K× the field extension L = K( m

√a) is cyclic and

its degree divides m. By Kummer theory the Galois group G of L|K isisomorphic to a subgroup of the m-th roots of unity; this isomorphism isgiven as follows: Choose α ∈ L such that

αm = a,

and define the faithful character χa on G by

χa(σ) = ασ−1 for σ ∈ G .

(This is independent of the choice of α as an m-th root of a.) Now, if theprime divisor P of K is relatively prime to a and m then P is unramified inL and its Frobenius automorphism σP satisfies

ασP ≡ αNP mod P ′

where P ′ is an extension of P to L. It follows

χa(σP ) ≡ αNP−1 = aNP−1m ≡

( aP

)mod P ′.

where the right hand side is the m-th Legendre symbol in K. Since the lefthand side and the right hand side are m-th roots of unity in K we conclude

χa(σP ) =( aP

)(4.1)

4I am indebted to the late Prof. Iyanaga for having translated Takagi’s review from theJapanese into German. The review appeared in Bull. Math. Phys. Soc. Japan I-2 (1927).


and hence by multiplicativity

χa(σB) =( aB

)(4.2)

for any divisor B which is relatively prime to a and m. In other words:

Suppose K contains the m-th roots of unity, so that the Jacobisymbol

(aB

)is defined for any divisor B relatively prime to m

and a. Let L = K( m√a). Then the Kummer character χa trans-

forms the Artin isomorphism B 7→ σB into the Jacobi homomor-phism B 7→

(aB

)from DS onto a subgroup of the m-th root of

unity. Consequently the kernel of the Jacobi homomorphism co-incides with the kernel of the Artin homomorphism, which is thegroup H ⊂ DS belonging to L as class field over K.

In particular it follows that the Jacobi symbol(aB

)depends only on the

class of B modulo the group H belonging to K( m√a) in the sense of Takagi’s

class field theory.

Let us discuss the special case when L|K is unramified, i.e., every primeP of K is unramified in L. Then we can take S to consist of the infiniteprimes only, hence DS = D is the full divisor group of K. The group Hbelonging to L is generated, firstly, by all divisor norms, and secondly bythose principal divisors (a) which are locally norms at all infinite primes Q.But the infinite primes Q are also supposed to be unramified in L whichimplies that LQ = KQ, hence every element a ∈ K× is trivially a local normat Q. Thus the second condition 2. above (page 22) is void. We concludethat H contains the group of all principal divisors of K, and consequentlyD/H is a factor group of the ordinary divisor class group C. Hence:

Suppose that K contains the m-th roots of unity. If K( m√a) is

unramified over K then the Jacobi symbol(aB

)depends only on

the class of B in the ordinary divisor class group C of K.

We have seen this statement as a consequence of Artin’s reciprocity law.It has turned out that precisely this statement is the essential ingredient inthe argument which shows that Artin’s reciprocity law implies the classicalreciprocity (3.16). As we have said above already, Hasse had pointed out


this fact to Artin (in a letter dated July 21, 1927). Hasse wrote that thisargument is a generalization of an argument which Furtwangler had used inhis earlier papers. When Artin had seen it he replied:

Vielen Dank fur den wirklich wundervollen Beweis von Furtwangler.

Many thanks for the truly wonderful proof by Furtwangler.

And Artin included Furtwangler’s “wonderful proof” in his paper [Art27].Let us briefly see what it is about:

4.1.1 Furtwangler’s wonderful proof

Let K be an algebraic number field containing the m-th roots of unity, leta, b ∈ K× be relatively prime to m and to each other, and suppose a ism-primary in K. In order to prove the classical reciprocity formula (3.16):(a

b

)=( ba

)it is convenient, Furtwangler says, to extend K to the field

K ′ = K(m√ab−1) . (4.3)

Now Furtwangler uses the relation between the Jacobi symbol in K andin K ′. Quite generally, for any finite extension K ′ of K and any divisor Bof K ′ which is relatively prime to a and m , it is easily checked from thedefinitions that ( a

N(B)

)K

=( aB

)K′

(4.4)

where N(B) denotes the relative norm of B to K. Apply this to the abovefield K ′ = K(

m√ab−1) and the divisor B = m

√(b). Note that by (4.3) the

principal divisor (ab−1) = (a)(b)−1 is an m-th power in K ′. Hence, since (a)and (b) are relatively prime to each other, we conclude that (a) and (b) eachare m-th powers of divisors in K ′ . Let, say,

(a) = Am and (b) = Bm ( in K ′). (4.5)


We have N(B) = Br where r = [K ′ : K], hence N(B)m/r = Bm = (b). (Notethat r divides m). Similarly N(A)m/r = (a) and it follows from (4.4):(a

b

)K

=( aB

)m/rK′

and( ba

)K

=( bA

)m/rK′

.

Therefore it suffices to prove in the field K ′:( aB

)K′

=( bA

)K′. (4.6)

In this way the problem is shifted from K to K ′. Now:

K ′( m√a) = K ′(

m√b) is unramified over K ′ . (4.7)

Indeed, the only primes of K ′ which possibly could ramify in K ′( m√a) are

the divisors of m and of a. But in K( m√a)|K the primes of m are unramified

since a is supposed to be m-primary in K.5 The property of being unramifiedis preserved if the base fieldK is extended to the overfieldK ′ hence the primesof m are unramified also in K ′( m

√a)|K ′. The primes of a are unramified in

K ′( m√a) since the principal divisor (a) is an m-th power in K ′.

In view of the unramifiedness (4.7) it follows from Artin’s reciprocity lawthat the Jacobi symbol

(aB

)depends only on the class of B in the ordinary

divisor class group of K ′. We have mentioned this on page 25. Similarly(bA

)depends on the class of A only. But A and B are in the same class sinceAB−1 is the principal divisor (

m√ab−1). Consequently, if X is a divisor in the

class of A and B such that X is relatively prime to m, a, b 6 then( aB

)=( aX

)=↑

( bX

)=( bA

)(in K ′) .

The equality sign which is marked by “ ↑ ” holds since a and b differ by afactor which is an m-th power in K ′, for

m√ab−1 ∈ K ′ by (4.3).

Indeed, isn’t this a wonderful proof as Artin had exclaimed?

To be sure, this proof works only if it is known that(aB

)K′

depends on theordinary divisor class of B only, as a consequence of the fact that K ′( m

√a) is

unramified. This fact was known to Artin as a consequence of his reciprocity

5Compare the definition of “primary” on page 19.6This guarantees that both

(aX

)and

(bX

)are defined.


law. But Furtwangler in [Fur09] did not have Artin’s reciprocity law at hisdisposal. So he had to prove that fact directly in the case m = ` which hewas concerned with. In fact, the main part of Furtwangler’s papers consistsof proving that fact with the methods which were available at that time.

4.2 Hilbert’s product formula

Hilbert, in the text of his 9-th Paris problem, did not explain in detail whathe meant by “reciprocity law”. What did he have in mind? Certainly notArtin’s reciprocity law; he was concerned with classical reciprocity. But didhe envisage a proof of (3.16) only? Or did he mean a more general reciprocity

law which gives an explicit expression of the inversion factor(ab

)(ba

)−1if

neither a nor b is primary ? Did he wish to include complementary laws ?

We find the answer in the text of the prize problem of the GottingenSociety which we have mentioned in section 2. There, after the statement ofthe problem we find some explanations, as follows:7

Bedeuten a, b irgend zwei ganze Zahlen des Korpers K und P irgend-ein Primideal in K so lasst sich das allgemeinste Reziprozitatsgesetzfur `-te Potenzreste im Zahlkorper K durch die Gleichung∏

P

(a, b

P

)= 1

darstellen. Hierin erstreckt sich das Produkt uber samtliche Prim-ideale P des Korpers K, und das Symbol

(a,bP

)bezeichnet eine in

geeigneter Weise zu definierende und durch die Zahlen a, b sowie dasPrimideal P eindeutig bestimmte Einheitswurzel.

Let a, b be any two integers of the field K and P any prime ideal ofK.8 Then the most general reciprocity law for `-th power residuesin the number field K is given by the equation∏

P

(a, b

P

)= 1 . (4.8)

7Our notation is different from Hilbert’s. Quite generally, when quoting mathematicaltext we use consistently our own notation which may differ from the various notations ofthe various authors.

8a, b are assumed to be nonzero.

4.2. HILBERT’S PRODUCT FORMULA 29

Here, the product is extended over all prime ideals of the field K,and the symbol

(a,bP

)denotes an `-th root of unity which is to be

defined suitably, uniquely determined by the numbers a, b and theprime ideal P .

And in Hilbert’s paper of 1899 [Hil99a] we find the following text:9

Es ist uberaus bemerkenswert, dass bei Anwendung unseres Symbolseine einzige Gleichung von so einfacher Bauart, wie es die Formel(4.8) ist, das Reziprozitatsgesetz fur einen beliebigen Zahlkorperzum Ausdruck bringt . . . die Formel nimmt Rucksicht auf die vie-len moglichen Falle, je nach der Realitat des Korpers und seinerKonjugierten; sie gilt, wie auch immer die Zerlegung der Zahl m imKorper ausfallen moge; sie enthalt alle Erganzungssatze; durch sieerscheint die exklusive Stellung der Zahl m und der in m aufgehen-den Primideale beseitigt; vor allem endlich gilt die namliche Formelunabhangig davon, ob die Klassenzahl des Korpers prim zu m oderdurch irgendeine Potenz einer in m aufgehenden Primzahl teilbarist.

It is highly remarkable that by using our symbol there is one sin-gle formula of such simple type like (4.8) which represents thereciprocity law in an arbitrary number field . . . the formula re-spects the many possible cases in regard to whether the field andits conjugates are real; the formula is valid independent of the de-composition of the number m in the field K; the formula containsall supplementary laws; the exceptional role of the number m andof the prime divisors of m is eliminated; but most notably thatformula holds regardless of whether the class number of the fieldis relatively prime to m or divisible by some power of a primenumber dividing m.

We conclude that Hilbert had meant to prove the product formula (4.8). Ashe explicitly mentioned, part of the problem was the proper definition of

9Here Hilbert referred to the quadratic case only which he dealt with in that paper.We have tampered a little with his text by replacing 2 by m. We believe that this wouldhave been approved by Hilbert since he propagated the generalization of his results toarbitrary exponents m.


the Hilbert norm symbol(a,bP

). This explains why Artin, immediately after

having obtained the reciprocity (3.16) for arbitrary m, wrote to Hasse in oneof his letters (dated July 17, 1927):

Man wird sich jetzt noch bemuhen mussen, daraus die HilbertscheFassung zu gewinnen . . . Einige oberflachliche Uberlegungen habenmir gezeigt, dass dies an keiner prinzipiellen Schwierigkeit scheitertund nur etwas langweilig ist . . . Hauptsachlich benotigt man es aberdoch nur in der vorigen Fassung.

One has still to go for the Hilbertian version . . . After some pre-liminary deliberations I got the impression that in principle therewill be no obstruction, only it will be somewhat boring . . . Butmainly one will need it in the above version.10

By “Hilbertian version” Artin means the product formula (4.8). In particularthe proper definition of the Hilbert symbol was required.

Hasse seemed to think that this task would not be that boring and hevolunteered to do the job if Artin would not like to do it himself. Artinseemed to be relieved, for he wrote on July 21, 1927:

Ich bin Ihnen im Gegenteil sehr dankbar . . . Ich kenne mich mit denNormenrestsymbolen doch bei weitem nicht so gut aus wie Sie.

On the contrary, I do appreciate this very much . . . I am not asfamiliar with the norm residue symbols as you are.

Here Artin refers to earlier papers of Hasse who had attempted to find adefinition of Hilbert norm symbols by purely local considerations, followingthe lead of Hensel (but only with partial success so far).

Now, what turned out to be the “proper” definition of the Hilbert symbolfor arbitrary m ?

Remember that the field K, in this connection, is supposed to containthe m-th roots of unity. As with the Legendre and the Jacobi symbol, weshould write

(a,bP

)m

or even(a,bP

)m,K

to indicate the number m and the field

10Artin means that the reciprocity law will be needed mainly in the form which he hadgiven to it, and not in the Hilbertian version.


K which is referred to. But again, in order to simplify notation we just write(a,bP

), thus considering m and K as being fixed in the course of the respective

consideration.

Let KP denote the P -adic completion of K and K×P its multiplicativegroup. The symbol

(a,bP

)is, by definition, a continuous bi-multiplicative

antisymmetric map from K×P × K×P into the group of m-th roots of unity(which are assumed to be contained in K) with the property that(

a, b

P

)= 1 iff b is a norm from KP ( m

√a) . (4.9)

These properties do not characterize the symbol completely. If P - m and mis odd then an explicit definition is given with the help of the Legendresymbol. Explcitly, Hasse defines(

a, b

P

):=

(a−vP (b) · bvP (a)

P

)· ε if P - m (4.10)

where vP (a), vP (b) denote the exponents with which P appears in a and brespectively. The additional factor ε = ±1 has to be inserted in order tosatisfy the norm condition 4.9. Explicitly, ε = 1 except when m is even andat the same time vP (a), vP (b) are both odd.11

Note that the Legendre symbol on the right hand side of (4.10) is definedsince the numerator is relatively prime to P , and P does not divide m.

This definition had been given by Hilbert in the case m = 2 and it wasquite clear that for arbitrary m the definition would have to be given asstated.

Remark: Actually, Hilbert himself did not work with the Henselian fieldKP . It seems that Hilbert had not realized the great potential of Hensel’sp-adic fields in number theory (although its underlying idea, namely theanalogy to local considerations in complex analysis, was quite familiar toHilbert who compared his product formula with Cauchy’s integral theo-rem).12 Hilbert referred to

(a,bP

)as a function defined on K× × K×, and

11This reflects the fact that for m even, if vP (b) = 1, i.e. if b is a prime element for Pthe norm of m

√b to K is −b and not b.

12In the year 1921 Courant, who was close to Hilbert, was heard saying that Hensel’sp-adic theory was an “unimportant sideway” (unfruchtbarer Seitenweg). See the Geleitwortin [Has75].


instead of (4.9) he said that modulo every power P r the residue of b shouldbe equal to the residue of the norm of some element from from K( m

√a). He

called(a,bP

)the “norm residue symbol”. It was Hensel [Hen22] who pointed

out that Hilbert’s condition for being a norm residue modulo every powerP r is equivalent to being a norm from the P -adic completion. Thereforethe name “norm symbol” has been substituted for Hilbert’s “norm residuesymbol”. Sometimes one says simply “Hilbert symbol”.

In [Has27] Hasse first showed that for primes P - m his symbol satisfies thenorm property 4.9. This turned out to be a little cumbersome but generated

no problem. His main task was the definition of(a,bQ

)for the critical primes

Q | m. His definition in [Has27] was given in an indirect and formal way.Hasse proved the following statement:

Product formula for the m-th Hilbert symbol.13 There isone and only one way to extend the local Hilbert norm symbol,which for primes P - m is given by (4.10), to the other primes

Q of K as a continuous bilinear antisymmetric symbol(a,bQ

)on

K×Q × K×Q with values in the m-th roots of unity, in such a way

that the product formula (4.8) holds for a, b ∈ K×. The normproperty (4.9) holds also for these Q.

The same statement for m = 2 had been proved by Hilbert, for m = `by Furtwangler and also by Takagi, and now by Hasse for arbitrary m. Butthis was by no means trivial. In each of these cases the essential basis of theproof was the relevant part of class field theory, and this had to be developedfor this purpose. The reciprocity law in the classical form (3.16) had to beproved first and from this the general statement above had to be deduced.

Today the above statement is considered almost trivial when class fieldtheory, including Artin’s reciprocity law, is seen in the framework of Cheval-ley’s ideles, by just using continuity arguments in the idele class group. Seesection 4.4 below. However, when it comes to explicit formulas, classicalreciprocity bears still some problems which are not solved by just referringto Hasse’s product formula statement above. For, that statement gives only

the existence and uniqueness of the Hilbert symbol(a,bQ

)for the critical

13Recall that the m-th roots of unity are supposed to be contained in K.


primes Q. It does not provide us with explicit formulas for the computation

of(a,bQ

). Perhaps this was the reason why Artin considered the proof of (4.8)

“somewhat boring”, as we have reported on page 30.

The necessity of explicit formulas for the critical primes is seen if we writethe product formula (4.8) in another way:

Assume that a, b are relatively prime to each other and to m. Then fromthe definition (4.10) it is seen that

∏P -m

(a, bP

)=(ab

)−1( ba

),

and hence the product formula (4.8) can be written as

(ab

)( ba

)−1

=∏Q|m

(a, bQ

), (4.11)

showing that the inversion factor can be expressed by the Hilbert symbolsfor the critical primes Q. If a or b are m-primary then the right hand sidevanishes in view of (3.16). In fact, each of the factors vanishes accordingto the very definition of “primary” given above, taking into considerationthe norm property (4.9) of the Hilbert symbol. But if neither a nor b are

m-primary then explicit formulas for the(a,bQ

)provide information for the

inversion factor, in those cases where such formulas are available.

In the case m = 2 and K = Q we have already stated a formula for theinversion factor; see (3.9). The two factors on the right hand side of (3.9)are14

(a, b2

)= (−1)

a−12· b−1

2 ,(a, bp∞

)= (−1)

sgn(a)−12

· sgn(b)−12 . (4.12)

Here, the symbol p∞ represents the infinite prime of the rational field Q.

At this point let us say some words about the so-called infinite primes ofa number field.

14Recall that in (3.9) a and b are assumed to be odd.


4.2.1 On infinite primes

By definition, a “prime” of a number field K is a valuation; more precisely,a class of equivalent valuations of K. There are two kinds of valuations, thenon-archimedean and the archimedean valuations. The non-archimedeansare in 1–1 correspondence to the prime ideals of the ring of integers in K.The archimedeans correspond to the isomorphisms of K into the complexfield, where two such isomorphisms belong to the same valuation if they differby the conjugate complex automorphism of C. The primes belonging to thearchimedean valuations are called the “infinite” primes of K. In contrast tothis the non-archimedeans are called “finite” primes.15 For an infinite primeQ of K its completion KQ is either R or C. If r1, r2 are the numbers of realresp. complex infinite primes then r1 + 2r2 = n, the degree of the field.

The Hilbert symbol for an infinite prime Q is defined for a, b ∈ K×Q asfollows:(

a, b

Q

)=

{(−1)

sgnQ(a)−1

2·sgnQ(b)−1

2 if Q is infinite real

1 if Q is infinite complex .(4.13)

This is in accordance with the above mentioned properties of Hilbert symbol,in particular with (4.9).

In product formulas like (4.8) it is understood that P ranges over allprimes of K, finite or infinite. Accordingly the infinite primes should beincluded in products like the right hand side of (4.11). Hence we should

have written more precisely∏

Q |mp∞

· · · , and we will do so if (and only if) the

infinite primes have to be explicitly mentioned. On the other hand, we seefrom (4.13) that an infinite complex prime does not give any contribution tothe product. If m > 2 then every infinite prime in K is complex since wehave required that K contains the m-th roots of unity. Thus if m > 2 onemay forget the infinite primes in product formulas like (4.11).

In the Hilbert text which we have cited above on page 29, it is said thatthe product formula (4.8) also comprises the so-called supplementary laws.

Let us explain this with the example of the quadratic symbol(

2p

)in Q which

15If we say “prime ideal” or “prime divisor” then we mean a non-archimedean prime. Ifwe just say “prime” then infinite primes are included.

4.3. HASSE’S NORM SYMBOL 35

we have given in (3.5). First we see from (4.10) and (4.12) that for odd primenumbers p 6= p ′:(2, p

p

)=(2

p

),(2, p

p ′

)= 1 ,

(2, p

∞

)= 1 .

Hence the product formula (4.8) reduces to(2

p

)(2, p

2

)= 1

In view of the norm property (4.9) of the Hilbert symbol we have if and onlyif p is a norm from Q2(

√2) and this is the case if and only if p ≡ ±1 mod 8;

otherwise(

2,p2

)= −1. This gives (3.5).

The point is that arguments of this kind, using Hilbert’s product formula,provide a method to find supplementary laws also for arbitrary m and forarbitrary number fields containing the m-th roots of unity. In the 1920s Hasseand Artin have published joint papers to determine supplementary laws form = ` an odd prime number, also for a power m = `ν , at least in the casewhen K = Q( m

√1) is the m-th cyclotomic field. Hasse’s Part II of his class

field report [Has30a] develops systematically explicit formulas for the critical

Hilbert symbols(a,bQ

)for Q | m, as far as possible.

4.3 Hasse’s norm symbol

We have mentioned above that Artin’s reciprocity law started a change ofparadigm in the realm of reciprocity. From then on the main emphasis wasput on class field theory. Takagi had shown this to be identical with thetheory of abelian extensions L|K of number fields [Tak20]. In Takagi’s theorythe base field K may be an arbitrary number field of finite degree, withoutany assumption about its roots of unity.

This had motivated Artin to look for some analogue to the Hilbert normsymbol in Takagi’s framework, i.e., without assuming that the base field Kcontains roots of unity. In this spirit he wrote to Hasse on July 21, 1927:

Eine ganz dumme Frage. Die σ-Formulierung des Rg. gilt in be-liebigen Korpern auch ohne irgend eine Einh.wurzel. Ist es nun


auch moglich, in beliebigen Korpern eine Art Normenrestsymbol zudefinieren? . . . Wahrscheinlich wird das nicht gehen. Aber warumnicht?

Quite a silly question. The σ-formulation of the reciprocity lawis valid in arbitrary fields, also in those without any root of unity.Now, is it possible to define also in arbitrary fields some kind ofnorm residue symbol? . . . Probably this will not work. But whynot?

When Artin says “arbitrary fields” then he means “arbitrary number fieldsK of finite degree”. By “σ-formulation” he means his new reciprocity law.This refers to a finite abelian extension L|K with Galois group G, and theArtin map A 7→ σA . This is a homomorphism from a certain divisor groupDS to G. See page 23.

Hasse regarded Artin’s question not so silly after all and he started towork on it. But it took him almost two years until he succeeded to realizeArtin’s suggestion. In the year 1929 he sent Artin a manuscript with hissolution, and Artin replied in a letter dated May 19, 1929:

Meinen herzlichsten Gluckwunsch zu der wundervollen Entdeckunguber das Normenrestsymbol. Das ist doch jetzt offenbar der wahreZusammenhang und man versteht erst jetzt die ganze Theorie. Esist alles so klar, dass nichts mehr zu bemerken ist.

My most heartfelt congratulations to your wonderful discovery ofthe norm residue symbol. This clearly is the true viewpoint, andonly now the whole theory can be understood properly. Everythingis so clear that there is nothing left to remark.

What were Hasse’s achievements that induced such enthusiastic commentsfrom Artin ?

The notation which Hasse used is somewhat different from Artin’s. In-stead of σP he writes (L|K

P). Thus the symbol (L|K

P) is defined for any prime

divisor P of K which is unramified in the abelian extension L, and it standsfor the Frobenius automorphism of P in the Galois group G of L|K. Clearly,Hasse had introduced this notation in order to blend with the notation ofthe Legendre symbol

(aP

)used in the classical problem of reciprocity.


Now, Hasse was able to define for all primes P of K a homomorphism

a 7→(a,L|KP

)from K×P to the the Galois group G of L|K which solves Artin’s

expectations. The method is similar to Hasse’s definition of the Hilbert sym-bol, namely: first he provided an explicit definition for the non-critical primesP which in the present case are those prime divisors which are unramifiedin L. Secondly he showed that, due to Artin’s reciprocity law, this defini-tion can uniquely be extended to the critical primes such that the productformula holds. As follows:

If P is a prime divisor of K which is unramified in L then the normsymbol is defined by putting(

a, L|KP

):=

(L|KP

)vP (a)

for a ∈ K×P . (4.14)

Recall that vP (a) is the exponent with which P appears in a. Since P is

unramified in L it is seen that the kernel of the map a 7→(a, L|KP

)consists

precisely of the norms from L×P , and its image is the decomposition group ofP in G. Observe that this definition does not require any root of unity to bein K. Hasse then proved the following statement:

Product formula for Hasse’s norm symbol. There is oneand only one way to extend the norm symbol which is originallydefined for the unramified prime divisors P by (4.14), to the other

primes Q of K, as a continuous homomorphism a 7→(a, L|KQ

)from K×Q to G, in such a way that the product formula holds:

∏all P

(a, L|KP

)= 1 for a ∈ K . (4.15)

The kernel of that homomorphism is the group of norms fromLQ|KQ and the image is the decomposition group of Q in G.

The product is to be extended over all primes P of K, including the infiniteprimes. Note that the “1” on the right hand of (4.15) side stands for the unitelement of the Galois group G.


This new symbol is called the “Hasse norm symbol”, in contrast to theHilbert norm symbol discussed in the foregoing section. Hasse’s paper ap-peared 1930 in Crelle’s Journal [Has30c].

If the m-th roots of unity are contained in K then the Hasse productformula (4.15) contains the Hilbert product formula (4.8). To see this onehas to verify from the definitions that(a, b

P

)m

=(a,K( m

√b)|K

P

)(4.16)

for every prime P of K. Because of the uniqueness statements in (4.15)and (4.8), it suffices to show it for prime divisors P which do not divide a, band m and hence are unramified in K( m

√b). Note that here the left hand

side is a Hilbert symbol, hence an m-th root of unity, whereas the right handside is a Hasse symbol, hence an element of the Galois group G of the cyclicextension K( m

√b)|K. By Kummer theory these groups can be identified, and

in this way the above relation is to be understood. More precisely, let χb bethe character of G defined by χb(σ) = ( m

√b) σ−1 for σ ∈ G. This yields an

injective homomorphism of G into the group of m-th roots of unity, and nowthe relation (4.16) is to be read as(a, b

P

)m

= χb

(a,K( m√b)|K)

P

)This is immediate from the definitions, recalling (4.2).

Consequently, the Hilbert norm symbols and their product formula havelost their importance, being a direct consequence of the product formula forthe Hasse symbols. We recall the enthusiastic exclamation of Hilbert whenhe had discovered his product formula as the source of the classic reciprocitylaw (see page 29). Nowadays this is superseded by Hasse’s product formula– since today the interest has shifted from classic reciprocity to class fieldtheory, as already mentioned. Instead of Hilbert we can now cite Artin’senthusiastic statement about the Hasse norm symbol (see page 36).

Hasse’s proof of (4.15) in his paper [Has30c] depends on Artin’s reci-procity law; he did not yet provide a new proof of Artin’s reciprocity law.Nevertheless this result of Hasse was a milestone in the evolvement of classfield theory. For he had succeeded to write Artin’s reciprocity law as a prod-uct formula – this is the first step on the way which leads to Chevalley’s


interpretation of class field theory in the framework of ideles. (For this seesection 4.4 below.) In view of this it is hard to understand that originallyArtin doubted that a product formula like (4.15) is possible; we have citedhis letter on page 35. Perhaps Artin had in mind a norm symbol which, likeHilbert’s, assumes its values in the m-th roots of unity, and indeed this is notpossible in a canonical way. But Hasse had the idea to construct a norm sym-bol with values in the Galois group, what Artin had called “σ-formulation”,and this is more natural and adequate to the problem.

Anyhow, after Artin had heard about Hasse’s result he realized imme-diately its importance and sent Hasse an enthusiastic letter; we have citedfrom this on page 36.

But in one aspect Artin‘s enthusiasm was too early. For, Hasse defined the

norm symbol(a, L|KQ

)for the critical ramified primes Q in a purely formal

way, similarly as in the former case of the Hilbert symbol(a, bQ

)m

for the

critical primes Q dividing m. Both these constructions were based on Artin’sreciprocity law which is a global affair. It took some more years until, in 1932,Hasse was able to follow a suggestion of Emmy Noether and give an explicit

definition of the norm symbol(a, L|KP

), based on the local theory of simple

algebras, for all P including the ramified primes. This allowed him to provethe product formula directly and hence to obtain a new proof of Artin’sreciprocity law [Has33]. See section ??.

4.3.1 Grunwald

In the year 1931 Hasse’s student Wilhelm Grunwald established an existencetheorem which in some sense can be considered as the inverse of Hasse’sproduct formula (4.15). While Hasse starts from an abelian extension L|Kand then constructs the local maps K×P → G satisfying the product formula,Grunwald starts from local maps satisfying the product formula and thenfinds a corresponding abelian extension L|K. The situation is as follows:

K an algebraic number field,

G a finite abelian group,

P ranges over all primes of K, including the infinite primes,

χP : K×P → G a continuous homomorphism (for each P ).


Without restriction it is assumed that G is generated by the images of the ho-momorphisms χP . In his thesis [Gru32] Grunwald’s result contains following:

Grunwald’s existence theorem. In the situation above, sup-pose that χP is unramified for almost all P .16 If the productformula ∏

P

χP (a) = 1 for a ∈ K× (4.17)

holds then there exists a unique abelian extension L|K and aunique isomorphism of G with the Galois group of L|K such that,if G is identified with that Galois group, each χP coincides withthe Hasse norm symbol:

χP (a) =

(a , L|KP

)for a ∈ K×P . (4.18)

The two results of Hasse and Grunwald correspond to the main theo-rems of the Takagi-Artin class field theory. Grunwald’s result is called theExistence Theorem of class field theory, establishing the existence (anduniqueness) of abelian extensions with given local norm symbols satisfyingthe product formula. Hasse’s theorem (page 37) says that inversely, everygiven abelian extension admits (uniquely) norm symbols satifying the prod-uct formula; therefore it is called the Inversion Theorem (Umkehrsatz ) ofclass field theory. The essential point is that both theorems had now beenformulated in terms of product formulas.

Already in the year 1926 Hasse had propagated to look more closely forproduct formulas as fundamental relations in number theory. In his paper[Has26b] he had observed for the rational number field Q and for the groupG = {±1}, that every product formula is connected to the quadratic reci-procity law. In other words, he had proved Grunwald’s theorem in this case.It is true that this special case is rather elementary. Hecke, when he hadread Hasse’s paper, had written to him on October 11, 1926: 17

16This means that χP is trivial on the P -adic unit group UP ⊂ K×P . It implies that inthe product (4.17) only finitely many factors 6= 1 appear.

17Hasse had been inspired by Hecke’s introduction of Grossencharacters and theirL-series; he wished to understand Hecke’s constructions by means of product formulas.


Ich verstehe die einzelnen Schritte und finde einzelne Teile auchhochst amusant, aber ich bin noch nicht dahintergekommen, wasdas alles eigentlich bedeutet.

I understand the various steps and I find certain parts very amus-ing but I do not yet see the intent of all this.

Hasse in his reply (dated October 18, 1926) said he wanted to show that

. . . der erste Fundamentalsatz (eind. Primzahlzerlegung) und der zweiteFundamentalsatz (quadr. Reziprozitatsgesetz) sich in ein- und dieselbeForm (Produkttheorem) setzen lassen und in dieser Art einzig sind.

. . . the first fundamental theorem (unique prime number decompo-sition) and the second fundamental theorem (quadratic reciprocitylaw) can be formulated in one and the same way (as a producttheorem) and that these are unique of their kind.

In other words, Hasse had wished to advocate the use of Hensel’s p-adicsin number theory. At that time, in the year 1926, the p-adics were not yetgenerally accepted as a useful tool in number theory. Hasse further explainedto Hecke that the product formula seems to be the key when switching fromlocal to global, although his (Hasse’s) ideas about this were still somewhatvague. We see that Hasse envisaged a local-global principle beyond the theoryof quadratic forms where he had discovered this principle in the first place(see, e.g., [Has24]). Of course the impact of Hasse’s idea would have beenmuch stronger if he had not limited himself to quadratic reciprocity in therational field. But in the year 1926 the general reciprocity law and theproduct formula for the norm symbol had not yet been established.

In 1930, when the relevant facts of reciprocity and class field theory hadbeen established, Hasse remembered his old paper of 1926 and now pro-posed to his student Grunwald the generalization to arbitrary number fields.Grunwald’s thesis corroborates what Hasse had envisaged in 1926.

From today’s viewpoint we see that, in fact, Hasse had represented, at least in the specialcase considered, Hecke’s Grossencharacters as homomorphisms of the idele class group.With the same aim he had later proposed to his student Grunwald to do the same inthe general situation, but the relation to Hecke’s Grossencharacters is not mentioned inGrunwald’s paper.


Remark: Actually, the situation which Grunwald considers in his thesis[Gru32] is more general than we have reported above. We have stated hisresult under the hypothesis that the χP are unramified for almost all P . Thisimplies that on the left hand side of (4.17) only finitely many terms are 6= 1,thus the product is essentially a finite product. Grunwald18 works undera weaker hypothesis, namely that the infinite product

∏P χP (a) converges

pointwise; this also implies that χP (a) = 1 for almost all P but he allows thatthere may be infinitely many χP which are ramified. In this more generalsituation Grunwald purports to show the existence of a sequence of abelianextensions Lν of K with group G such that the corresponding norm symbols(a, LνP

)converge pointwise to χP (a) (for ν →∞).

This more general result has not been followed up in the literature, notonly because there is an error in Grunwald’s arguments19 (which can bestraightened up though) but mainly since the hypothesis of pointwise con-vergence does not correspond to the natural topology of the idele group (forthis see below page 44). Perhaps this is the reason why Grunwald’s paper hadnot become widely known and his existence theorem had not been connectedwith his name.

Today the name of Grunwald is remembered in the mathematical com-munity mainly because of the said error. This error is connected with alemma which is also an existence theorem but refers to finitely many primesonly, without requiring the validity of a product formula. Namely, Grunwaldconsiders finite sets Sν of primes of K which grow with ν and will exhaustthe set of all primes of K. (For reasons of simplicity G is assumed to be acyclic group.) The erroneous lemma claims for a fixed set Sν the following.

Assume G is cyclic. Let Sν be a finite set of primes P of Kand χP : K×P → G continuous homomorphisms for each P ∈ Sν,such that G is generated by their images. Then there exists (notuniquely) a cyclic extension Lν |K with group G such that theχP (a) for the finitely many P ∈ Sν coincide with the Hasse normsymbols

(a, LνP

).

18And also Hasse in his earlier publication [Has26b] mentioned above.19This error does not affect the validity of Grunwald’s existence theorem which we have

stated above on page 40.

4.4. CHEVALLEY’S IDELES 43

Grunwald did not state this explicitly in his thesis [Gru32] but implicitly heproved and used it in the course of his arguments. When Hasse read the thesishe realized that he could well use this existence lemma in his proof of cyclicityof simple algebras over a number field.20 Therefore he asked Grunwald towrite another paper as a source for reference, where this existence lemma isexplicitly stated and the proof given. Grunwald did so and submitted thissecond paper to Hasse on January 31, 1931 [Gru33]. But this second paperwas not entirely self-contained since for certain details he still referred to histhesis.

The story is well known:21 It took 14 years until the error in Grunwald’spaper was detected by Sh. Wang, a Ph.D. student of Artin. This happenedin Artin’s seminar in Princeton 1947. In fact, Wang constructed a counterexample to Grunwald’s existence lemma. The lemma was corrected by Wanghimself [Wan50] and the corrected version is now called the Grunwald-Wangtheorem. Corrections were also given by Hasse [Has50] and by Artin-Tate[AT90].22

But, as said above, this error of Grunwald does not affect his proof of theexistence theorem above.

4.4 Chevalley’s ideles

In the year 1936 there appeared a paper by Claude Chevalley [Che36] in whichwere introduced what today are called the “ideles” of a number field K. Thefirst known document where this notion is mentioned is a letter of Chevalleyto Hasse dated already a year earlier, on June 20, 1935. Originally Chevalleydid not yet use the word “ideles”; he used elements ideaux (ideal elements)instead. Here is what he wrote to Hasse:

Soit K un corps fini de nombres algebriques. Considerons les diversdiviseurs premiers P de K et les corps locaux correspondants KP .Appellons elements ideaux (e.i.) of K les systemes comportant un

20That result is usually referred to as Hasse-Brauer-Noether Theorem.21See, e.g., [Roq05].22As it had turned out, both these corrections had to be corrected again. See [Mor11]

and [GJ07].


element aP (le P -composant) dans chaque KP , ces composantesetant choisis de maniere a ce que seuls un nombre fini de composantsne soient pas des unites.

Let K be a finite algebraic number field. Consider the variousprimes P of K and the corresponding local fields KP . We callelements ideaux (e.i.) of K those systems which consist of anelement aP (the P -component) in every KP , chosen such thatonly finitely many of them are not units.23

These elements ideaux form a multiplicative subgroup of the direct product∏P K

×P . Let JK denote this subgroup. Today this is called the group of

“ideles” of K. The non-zero elements a ∈ K× are diagonally embedded intoJK ; they are called the “principal ideles” and the factor group CK = JK/K

×

is the group of “idele classes” of K.

Consider a system {χP} of continuous homomorphisms χP : K×P → Ginto a finite group G. Suppose that for almost all P the χP are unramified,as it was supposed, e.g., in Grunwald’s existence theorem which we stated onpage 40. Then we obtain a map χ : JK → G by defining for a = {aP} ∈ JK

χ(a) =∏P

χP (aP ) . (4.19)

For almost all P we have χP (aP ) = 1 and hence the right hand side is a finiteproduct. If the χP satisfy the product formula then this can now be writtenas χ(a) = 1 for a ∈ K×. In other words:

The product formula (4.17) signifies that the character χ definedby (4.19) is a homomorphism from the idele class group CK =JK/K

× into G.

Chevalley explains in his letter to Hasse that there is a natural way tointroduce a topology in JK such that JK becomes a topological group andthe continuous homomorphisms χ : JK → G into a finite group G are givenprecisely in the above manner. In other words: By definition, a homomor-phism χ : JK → G is continuous in Chevalley’s idele topology if and only if:

23Chevalley did not explicitly mention that the infinite primes P are to be included inthis construction. Also, the aP have to be 6= 0.


1. Every component χP : K×P → G is P -adically continuous ,

2. χP is unramified for almost all P .

In this topology K× is a discrete subgroup of JK . The idele class groupCK = JK/K

× inherits its topology from JK .24

Hasse’s construction of the norm symbols(a, L|KP

)(see page 37) can now

be formulated as follows:

Let L|K be a finite abelian extension with Galois group G. Thenthere is a canonical surjective homomorphism from the idele classgroup CK onto G such that its P -components for unramified fi-nite P are given by (4.14). This homomorphism is continuous inthe idele topology and its kernel is the norm group NL|KCL.

This is the idelic version of Artin’s original reciprocity law of 1927.

If a = {aP} is an idele in JK we denote its image under that homomor-phism with (a, L|K) so that

(a, L|K) =∏P

(aP , L|KP

)for a ∈ JK ,

(a, L|K) = 1 for a ∈ K× .

The existence theorem of Grunwald which we have discussed on page 40can now be stated as follows:

Let χ be any continuous homomorphism of the idele class group CKwhose image χ(CK) = G is a finite group. Then there exists aunique finite abelian extension L|K and a unique isomorphismof G with the Galois group of L|K such that, after identifying Gwith this Galois group we have

χ(a) = (a, L|K) for a ∈ CK .

24This topology of CK is not Hausdorff. We shall come to this question later.


This is the idelic version of the existence theorem of class field theory.We see that Chevalley’s introduction of ideles and idele classes permits abeautiful view of the structure of class field theory. We also see that the ideahad been prepared by Hasse and Grunwald when they stressed the role ofproduct formulas in class field theory. But they did not take the last step ofcombining all the local components K×P into one group JK . This step wasnow done by Chevalley.

4.4.1 Chevalley’s encyclopedia article

But why did Chevalley explain this to Hasse in a long letter (five typewrittenpages) and did not wait for his paper to appear, when he could just send hima reprint?

The immediate motive for Chevalley’s letter to Hasse had not been theannouncement of his paper but to discuss an article on class field theorywhich he was preparing for the German Encyclopedia of Mathematics25 ofwhich Hasse was one of the editors.

The project of the German Encyclopedia of Mathematics is quite old; werefer to the survey of Tobies [Tob94]. Its first volume, dealing with arithmeticand algebra, had appeared in the year 1904. Now, at the beginning of the1930s, this was quite outdated and it was widely felt that a new edition ofthis volume was needed. Finally, in 1934 Hasse and Hecke were nominatedas the chief editors of the 2nd edition of volume I and they started to enlistauthors for the various articles.26

Hasse had proposed the name of Chevalley as the author for the articleon class field theory. He believed that now the young Chevalley (he was 25years of age in 1934) was the best author for this job. For, Chevalley hadpublished in his thesis [Che33] a comprehensive report on the new foundation

25Encyklopadie der mathematischen Wissenschaften unter Einschluß ihrer Anwendun-gen.

26According to Tobies [Tob94] they insisted that the project should be operated on atruly international scale, as far as the authors of the various contributions were concerned.The best experts world wide should be asked for cooperation, including German emigrantsand Jewish authors. However in 1939 the publisher (Teubner) decided not to print arti-cles from those authors. From the correspondence Hasse–Hecke it can be seen that theyopposed this decision but they were unable to have it changed since apparently Teubnerwas put under pressure by the Nazi government.


of class field theory based on Hasse’s local class field theory. This includedalso great simplifications of proofs which were due to Chevalley himself and toHerbrand, Artin, Hasse. This work was regarded as an update of Hasse’s classfield report [Has26a] which formerly had served a number of mathematiciansas an introduction and valuable source for class field theory.

On Hasse’s inquiry Chevalley had agreed to write a survey article for theEncyclopedia. His letter of June 25, 1935 from which we have cited aboveshows that his idea about the introduction of ideles had evolved during hiswork composing that article. For the letter starts as follows:

Pendant la redaction du chapitre “Theorie du corps de classes” pourl’Encyclopedie allemande, il m’est venu quelques idees sur l’expositionde la theorie, dont je vous envoie un resume . . .

While composing the chapter “Class Field Theory” for the Ger-man Encyclopedia, I got some ideas about the exposition of thetheory, of which I would like to send you a resume . . .

He continues to explain the notion of “elements ideaux”, as we havereported above. At the end of the letter he writes that he would like to presentclass field theory using these new ideas, but this would perhaps require torewrite much of what he had already done, and then he would not be ableto keep the deadline for submission of the manuscript. Therefore he askedHasse how he should proceed in this situation.

Hasse was quite excited about Chevalley’s idea of elements ideaux. Hereplied on June 28, 1935:

Die Art, wie Sie jetzt die Klassenkorpertheorie aufziehen, entsprichtvollkommen einem lange von mir gehegten Wunsch, namlich dieZusammensetzung der lokalen Klassenkorpercharakterisierungen zuder Charakterisierung im Grossen so zu machen, dass man aus demfertigen Resultat die lokalen Bestandteile wieder herauslesen kann.Hierzu ist eben die Takagische Gruppe ungeeignet, weil der Zusam-menhang zwischen Artinsymbol und Normensymbol zu kompliziertist.

The way how you now present class field theory complies perfectlywith my long standing desire, namely the synthesis of local and


global class field theory in such a way that from the final resultsone may recover again the local components. The Takagi group isnot suitable for this purpose since the relationship between Artinsymbol and norm symbol is too complicated.

By Artin symbol Hasse means Artin’s σP =(L|KP

), and by norm symbol he

means what we have called the Hasse norm symbol(a, L|KP

); see page 37.

And Hasse granted Chevalley any time which he should need to put classfield theory into his new setting. Hasse also recommended to look at his old1926 paper [Has26b] and also to Grunwald’s paper [Gru32] in ordet to seewhether Chevalley could use it, but it turned out that the latter had alreadyseen them.

Chevalley’s encyclopedia report has never appeared. The reasons wereon the one hand that at a certain stage he wished to have the manuscriptreturned since he had new proofs without use of analysis. On the other hand,when his new manuscript finally was received it had to be translated fromFrench into German27 which required some time. And then the outbreakof world war II prevented the printing. There is a letter of Hasse datedOctober 19, 1948, i.e., after war’s end, in which he asks Chevalley whetherthe manuscript could now be printed or perhaps Chevalley would like torewrite some parts? We do not know Chevalley’s answer. We have tried atseveral places to find this manuscript of Chevalley, either in its translatedversion or the original French version, but without success.

4.4.2 Chevalley’s final paper

Since Chevalley’s encyclopedia report did not contain full proofs he intendedto write a separate paper on his new view of class field theory, including hisnew proofs without analysis. He intended to submit this to Crelle’s Journal,as he wrote to Hasse on January 15, 1938:

Quant au memoire dont vous parlez, sur la theorie du corps declasses, il y’a longtemps que je desire l’ecrire, et j’avais justementl’intention de vous le proposer pour le Journal de Crelle.

27At that time English was not generally established as the lingua franca of science. Thetranslation was assigned to M. Eichler who at that time was Hasse’s assistant in Gottingen.


Concerning the paper on class field theory which you mentioned,I wished for a long time to write it up, and indeed I intended tosubmit it to you for Crelle’s Journal.

But he added that he did not yet find the time to complete the paper sincehe was occupied with other duties, and moreover

. . . il y’a toujours ce traite d’analyse collectif qui exige un travailconsiderable.

. . . there is still the collective treatment of analysis which requiresconsiderable work.

Of course he means his participation on the collective Bourbaki project.

Chevalley told Hasse that in October 1938 he intends to go to Princetonwhere he had received an invitation from the Institute for Advanced Study.But just before leaving for USA he presented a talk, on Hasse’s invitation,about his new class field theory at the annual meeting of the German Math-ematical Society (DMV) in Baden-Baden on September 12, 1938. But hispaper on class field theory appeared not in Crelle’s Journal (probably becauseof war-time difficulties) but in the Annals of Mathematics instead [Che40].This was the first paper where class field theory had been developed fromscratch in the idelic framework. As mentioned above already, a particularfeature of the proof was that it got along without use of analysis, as we havealready mentioned.

Another particular turn in Chevalley’s 1940 paper was that he did notrestrict his considerations to finite abelian extensions but he worked fromthe start with the maximal abelian extension Kab of a number field K. ItsGalois group Gab is compact with the Krull topology. Chevalley consideredits character group Gab and showed that it is canonically isomorphic to thegroup of continuous characters of the idele class group CK . Then he usedPontrjagin’s duality theorem and obtained a canonical isomorphism of Gab

to, well, not to CK but to the maximal compact factor group of CK . (Weshall discuss later the structure of CK as a topological group and its maximalcompact factor group.)

From today’s viewpoint, the transition to the full abelian extension Kab

from the various finite subextensions does not really require much work. We


are used to the concepts of injective and projective limits, and the class fieldtheory of Kab|K is nothing more than the simultaneous considerations ofthe finite abelian extensions L|K, together with the observation that theobjects to be discussed satisfy the relevant functional properties. In fact,Chevalley’s invention of the notion of idele which he did in his 1936 paper[Che36], was motivated to find the proper object which enjoys the relevantfunctional properties with respect to the change of the abelian field extensionL|K.

But we have to take into consideration that in 1935, when

This result contained all of the ordinary class field theory which con-sidered only the finite abelian extensions. Specifically, the finite abelianextensions L|K are in 1-1 correspondence with the closed subgroups of finiteindex H ⊂ CK such that H = NCL is the group of norms from CL; thefactor group CK/H is canonically isomorphic to the Galois group G of L|Kvia the Hasse norm map a 7→ (a, L|K) (see page 45). Chevalley’s emphasishere is that all class field theory for the various finite abelian extensions iscombined in CK .

Actually, the Galois theory of infinite algebraic field extensions was notnew. It had been started in the year 1928 by Krull [Kru28] who observed thatthe Galois group of an infinite Galois extension carries naturally a compacttopology. The ramification theory of infinite Galois extensions of numberfields had been cleared up by Herbrand and Ostrowski in various papers inthe early 1930s. In fact,

Concerning Chevalley’s 1936 paper which we have mentioned at the be-ginning of this section: The original motivation for that paper was not anew foundation of class field theory, but Chevalley wished to extend classfield theory to infinite abelian extensions, and for this purpose he needed hiselements ideaux. He had found out that “the Takagi group is not suitable”for this purpose (same as Hasse’s expression which we have cited above).

4.4.3 On the name “idele”

Chevalley’s paper was reviewed in the two reviewing journals which existed atthat time: first the Zentralblatt fur Mathematik and secondly the Jahrbuchuber die Fortschritte der Mathematik. The reviewers were Grunwald and


Hasse respectively. Both reviews were quite long and they mirror the greatenthusiastic interest which Chevalley’s discovery had found among the Ger-man number theory group around Hasse.

While Chevalley in his paper had still used the expression “element ideal”Grunwald and Hasse in their reviews proposed to substitute “Idel” instead.This artificial word in German language is obtained from “Ideal”, whichrepresented a well defined notion in the mathematics of the time, by removingthe letter “a”. This should indicate that the new concept is in some sense ageneralization or rather a refinement of the old. Hasse explaines this in hisreview as follows:

Der Begriff des Ideals entspringt aus der Nichterfullbarkeit der For-derung, eine Zahl aus K zu finden, die fur alle Primstellen P von Kvorgeschriebene Ordnungszahlen hat, wobei fur fast alle P die Ord-nungszahl 0 vorgeschrieben ist. Verscharft man die durch die Ord-nungszahlen ausgedruckten groben P -adischen Annaherungsvorschrif-ten durch Vorschreiben der gesamten P -adischen Entwicklungen, soentspringt der vom Verf. neu eingefuhrte Begriff des idealen Ele-ments (Idels).

The notion of ideal has its source in the fact that the problem tofind a number in K with prescribed multiplicities at all prime di-visors P (whereby for almost all P the multiplicity 0 is prescribed)is not always solvable. The prescription of the multiplicities repe-sents a coarse P -adic approximation requirement. If this require-ment is tightened by prescribing the complete P -adic expansions,there arises the author’s new notion of ideal element (idele).

Grunwald’s review appeared earlier than Hasse’s and it would appearthat it was he who had first proposed the name “Idel”. But Grunwald hadbeen a Ph.D. student of Hasse and they were in close contact28, hence webelieve that it was Hasse who had proposed this name. This name was lateraccepted by Chevalley who wrote it in French as “idele”. In English this wastaken over without accent: “idele”.

28The friendly contact between Hasse and Grunwald lasted life-long.

Chapter 5

The Highlights

5.1 The Local-Global Principle

There is another result in Furtwangler’s early papers which later becameimportant and which therefore we wish to discuss here. This is the followingtheorem:

Local-global principle for norms. Let L|K be a cyclic exten-sion of number fields, of prime degree [L : K] = `. If an elementa ∈ K× is locally a norm from L×P for each prime P of K (in-cluding the infinite primes), then a is a norm from L.

Furtwangler himself did not formulate this theorem as such but Hasse inhis class field report extracted it and its proof from Furtwangler’s papers;see [Has30a] §8. At the same time Hasse conjectured that this local-globalprinciple would generalize to arbitrary abelian extensions L|K. However thisturned out not to be the case. Furtwangler’s local-global principle for normsholds for cyclic extensions, but not generally for abelian extensions.

Today we prefer to look at the problem with the eyes of cohomology. LetG be the Galois group of L|K. For any G-module X the cohomology groupH0(G,X) is defined as the group of G-invariant elements in X modulo thesubgroup of G-norms. The inclusion L× ⊂ JL leads naturally to a map

H0(G,L×)→ H0(G, JL). (5.1)

53

54 CHAPTER 5. THE HIGHLIGHTS

Furtwangler’s norm principle says that this map is injective provided G iscyclic of prime order `.

It is well known that for a cyclic group with fixed generator there is acanonical isomorphism H0(G,X) = H2(G,X) for any G-module X. Here,H2(G,X) is defined to be the group of so-called G-factor systems in X mod-ulo the split factor systems. Hence for cyclic G the map (5.1) can be inter-preted as

H2(G,L×)→ H2(G, JL). (5.2)

This map can be handled in the following way.

Recall that H1(G,X) is defined as the kernel of the G-norm N : X → Xmodulo the group generated by the subgroups Xσ−1 for σ ∈ G. It is wellknown that H1(G,L×) = 1 . This result is often referred to as “Hilbert’sTheorem 90”. 1 We also have for the idele group H1(G, JL) = 1 which inessence is Hilbert’s Theorem 90 for the localizations LP |KP for the primesP of K. Now consider the exact sequence for ideles:

1→ L× → JL → CL → 1

where CL = JL/L× is the group of idele classes. This yields an exact coho-

mology sequence

1 = H1(G, JL)→ H1(G,CL)→ H2(G,L×)→ H2(G, JL) . (5.3)

Consequently, the injectivity of the map H2(G,L×) → H2(G, JL) is equiva-lent to

H1(G,CL) = 1 . (5.4)

We conclude:

Assume G cyclic. Then the Furtwangler local-global principle fornorms is equivalent to H1(G, JL) = 1.

Indeed, if one follows the original proof of Furtwangler in the case whenG is of prime order (the proof has been reproduced by Hasse [Has30a]) then

1However Hilbert in his Zahlbericht [Hil97] proved this for cyclic extensions L|K only.A proof for arbitrary Galois extensions, using non-commutative algebra, is given by EmmyNoether in [Noe32]. There she cites Speiser [Spe19] for an earlier proof.

5.2. SIMPLE ALGEBRAS 55

one sees that this proof rests essentially on the the vanishing of H1(G,CL)which has to be proved beforehand. Of course, at those times neither thenotion of idele nor the formalism of cohomology had been in use. But ineffect, what was done at that time were essentially the steps which today wewould interpret as the proof of H1(G.CL) = 1.

The statement H1(G,CL) = 1 has obtained the name “principal genustheorem” because its historic roots go far back to Gauss’ genus theory forquadratic forms. The elements of the factor group of CL modulo the kernelof the norm are called “genera”. Accordingly the kernel of the norm is the“principal genus”. The principal genus theorem says that this coincides withthe group Cσ−1

L where σ is a generator of G.

The version with Chevalley’s ideles seems to be the most lucid one. Butthis does not mean that the proof in this version becomes easier.

In all versions known so far the principal genus theorem appears notdirectly but only as a fall out of the main statement of class field theory,namely that the order of the norm factor group h = (CK : NCL) equals thedegree of the field extension n = [L : K]. This proof was usually dividedinto two parts: first h ≤ n and secondly h ≥ n. While proving the secondinequality it turns out that even h ≥ n · γ where γ denotes the order ofH1(G,CL) (or H1 of a certain divisor class group). Taken together this yieldsH1(G,CL) = 1.

After Artin’s proof of his reciprocity law and Hasse’s interpretation of itby product formulas, Hasse brought Furtwangler’s norm principle forwardagain, and he asked for its generalization. One of the reasons was that thisquestion came up in the theory of simple algebras.

5.2 Simple algebras

The story is a follows:

From February 26 to March 1, 1931 Hasse in Marburg had organized asmall workshop. He wished to advance the theory of simple algebras withthe ultimate aim of proving that every simple algebra over a number field iscyclic. He was convinced that Furtwangler’s norm principle was an importanttool in this endeavor. At the Marburg workshop he delivered a lecture on


cyclic algebras but he was not yet able to generalize Furtwangler’s theoremto the case of arbitrary cyclic extensions.

But a week after the workshop Hasse sent a short announcement, datedMarch 6, 1931, to every participant:

Habe soeben den fraglichen Normensatz fur zyklische Relativkorperbewiesen, und mehr braucht man fur die Theorie der zyklischen Di-visionsalgebren nicht.

Just now I have proved the norm theorem in question for rela-tively cyclic fields, and more is not needed for the theory of cyclicdivision algebras.

When Hasse mentions “relatively cyclic fields” then he means cyclic ex-tensions L|K of arbitrary degree, not necessarily prime. His proof in [Has31]uses induction, thus reducing his norm theorem to Furtwangler’s case ofprime degree. But at the same time he presented a counterexample to hisconjecture that the norm theorem would be valid also for arbitrary abelianextensions. In fact, he noticed for L = Q(

√−39,

√−3) that 3 is not a norm

from L but locally it is a norm everywhere.

Let us recall how the norm problem affects the theory of cyclic algebras:Suppose that L|K is a cyclic field extension of degree n, and let σ be a gen-erator of its Galois group. Let a ∈ K. Consider the K-algebra A containingL and an element u with the defining relations :

un = a , xu = uxσ for x ∈ L .

This is a simple algebra over K, of dimension n2. If a is a norm from L thenA splits, i.e., A is a full matrix algebra over K, and conversely. Hence theHasse-Furtwangler result can also be stated in the following way:

Local-global principle for cyclic algebras. Let A be a cyclicalgebra over a number field. If AP := A ⊗KP splits locally overKP for every prime P of K (including the infinite primes) thenA splits over K.

Hasse included this result in his manuscript on cyclic algebras over num-ber fields, which he submitted in May 1931 to the Transactions of the AMS


for publication [Has32b]. This is the paper where the Hasse invariants ofsimple algebras are defined which later would become important in Hasse’snew proof of Artin’s reciprocity law.2 See section ??.

As said above already, Hasse conjectured that every simple algebra overa number field K is cyclic.3 If this would be so, then his above local-globalprinciple would apply to all simple algebras over K. However, in order toprove his conjecture Hasse needed to know beforehand that there is a local-global principle for all simple algebras, irrespective of whether it is knownalready that they are cyclic. And only with the help of this he was able todeduce that, in fact, every simple algebra over a number field is cyclic. See,e.g., [Roq05].

In order to follow this idea it would be necessary to find a suitable gener-alization of the Furtwangler-Hasse local-global principle, which is applicableto all Galois extensions L|K. But how to find it?

The crucial idea came from Emmy Noether. In the spring of 1931 Hassevisited her in Gottingen and, as it was her custom, she took him on anextended hiking tour through the hills around the town. In the course ofthat tour Hasse told her about his counter example. Upon this she advancedthe suggestion that the correct version of the general local-global principlemight be in terms of factor systems instead of norms. This is precisely whatwe have seen in (5.3). It is remarkable that she immediately understood thesituation and came up with this suggestion, although at that time algebraiccohomology had not yet been developed. But she knew that simple algebras,cyclic or not, can be characterized by factor systems, more precisely, byequivalence classes of factor systems. Emmy Noether had developed hertheory of factor systems representing simple algebras in her Gottingen lecture1929. It was first published by Hasse (with Noether’s permission) in hisTransactions paper mentioned above [Has32b].

Hasse followed Noether’s suggestion and accordingly tried to prove theinjectivity of H2(G,L×) → H2(G, JL), this time for arbitrary Galois ex-tensions L|K, not necessarily cyclic. In view of (5.3) this is equivalent to

2For some reason Hasse had never received the proof sheets of this paper, hence thereremain quite a number of misprints. Consequentrly, while reading the paper one shouldat the same time consult [Has32a] where the misprints are corrected.

3Sometime in the literature this conjecture is attributed to Dickson.


H1(G,CL) = 1, i.e., to the principal genus theorem in this generality. Wehave reported in [Roq05] on the conceptual difficulties which Hasse had toovercome until finally a very simple proof was found, in cooperation withRichard Brauer and Emmy Noether. This happened on November 9, 1931.The contribution of Brauer consisted of providing a Sylow argument whichreduced the problem to the case of p-groups. But p-groups are solvable.Noether’s contribution consisted of providing a simple induction argumentwhich reduced the problem from solvable groups to groups of prime order,hence to Furtwangler’s case.

In this way the famous local global principle was obtained:

Local-global principle for simple algebras. Let A be a simplealgebra over a number field K. If AP := A⊗KP splits locally overKP for every prime P of K (including the infinite primes) thenA splits over K.

Let us say again that factually this is the same as the above local-globalprinciple for cyclic algebras since after all, as a consequence of this, Hassehad shown that every simple algebra over a number field is indeed cyclic.4

We remark that the Brauer-Noether reductions to Furtwangler’s case ofprime degree do not involve the principal genus theorem H1(G, JL) = 1.For, these reductions work with H2 only which represents the Brauer groupof simple algebras. Thus one can say that the above local-global principle forsimple algebras is a direct and straightforward consequence of the principalgenus theorem in the case of prime degree n = `, and this in turn is a fall-outof the proof of the main theorem h = n of class field theory in the case n = `.(See page 55.)

There have been attempts to find a direct proof of the local-global prin-ciple for simple algebras without recurrence to the details of the proofs inclass field theory but so far without success. It has been possible, however,

4Well, Hasse had used in his original proof the erroneous existence lemma of Grunwaldwhich we have stated on page 42. Hence his proof was not complete until Wang hadcorrected Grunwald’s existence lemma in 1947 and shown that the modified lemma sufficesfor the use in Hasse’s argument. Therefore, the name of Wang should perhaps also bementioned in connection with the cyclicity theorem for algebras over number fields.


to greatly simplify the details of proof. This has been achieved by Herbrand5

and Chevalley in the year 1931. Artin wrote to Hasse on June 16, 1931:

Begeistert bin ich uber die neuen ungeheuren Vereinfachungen derKlassenkorpertheorie, die von Herbrand und Chevalley stammen. Manbraucht jetzt so gut wie gar keine schlimmen Rechnungen mehr, auchkeine trinomischen Gleichungen wie F.K. Schmidt. Da ich an eini-gen kleinen Punkten auch beteiligt bin, mochte ich Ihnen ganz kurzdaruber schreiben . . .

I am enthusiastic about the tremendous simplifactions of classfield theory due to Herbrand and Chevalley. Almost no dreadfulcalculations are required anymore, nor any trinomial equations asF. K. Schmidt. Since I am involved in a few small points I wouldlike to write you briefly about it . . .

These simplifications were concerned with the proof of the main theorem ofclass field theory and as such they became relevant also for the proof of theprincipal genus theorem. It turned out that the Chevalley-Herbrand proofworked at once for arbitrary cyclic extensions; the reduction to Furtwangler’scase of prime degree was not necessary any more. For the details we refer toChevalley’s thesis [Che33].

Later, Emmy Noether has formulated the principal genus theorem forarbitrary extensions L|K in the framework of non-commutative algebras[Noe33]. It appears that her aim was a direct proof without going throughthe proofs of class field theory. But she did not achieve this. She wrote:

Der Beweis ergibt sich aus dem oben genannten Satz uber zerfal-lende Algebren durch rein algebraisch-arithmetische Uberlegungen,wahrend in ersterem Satz der Normensatz im zyklischen Fall – esgenugt Primzahlgrad – als transzendenter Kern steckt.

The proof 6 uses the above mentioned theorem on split algebras 7

with purely algebraic-arithmetic arguments, whereas that theorem

5Jacques Herbrand was 23 years of age when he died on August 4, 1931 on an alpinetour. Before he had spent 10 months as a Rockefeller fellow at the German universitiesBerlin (von Neumann), Hamburg (Artin) and Gottingen (Emmy Noether). He also hadvisited Hasse in Marburg on the occasion of Hasse’s workshop about cyclic algebras.

6Of the principal genus theorem.7Here Noether refers to the local-global principle for algebras.


is based on the norm theorem in the cyclic case – it suffices ofprime degree – which uses transcendental methods.

Here Noether refers to the fact that a certain part of the proof of themain theorem of class field theory required methods from analytic numbertheory 8, whereas she strives to find a purely algebraic proof. Now, in thelater development a purely algebraic proof has been found, by Chevalley[Che40]. But still the situation prevails that the local-global principle foralgebras cannot yet be proved without diving into the details of class fieldtheory – if only in the case of prime degree.

8This part is the proof of the first inequality h ≤ n in the notation explained above onpage 55.

Chapter 6

Local theory

[Sch30]

6.1 Local class field theory

Let L|K be an abelian extension of number fields and P a prime of K. In sec-

tion 4.3 we have introduced Hasse’s norm symbol(a, L|KP

)as a map from the

completion K×P to the Galois group G of L|K whose image is the decomposi-tion group GP and whose kernel is the norm group NL×P . If P is unramifiedin L then, by definition, this map sends a ∈ K×P to the vP (a)-th power of the

Frobenius automorphism(L|KP

). See (4.14). Hasse’s achievement was the

definition of the norm symbol for the ramified primes. He did this on thebasis of Artin’s reciprocity law, as we have explained in section 4.3, page 37.

After having established the relevant properties of his norm symbol, Hasserealized that these properties could be regarded as the main theorems of alocal class field theory which looks quite similar to the global class field theoryof Takagi-Artin.1 He wrote a second paper [Has30b] where he explains thisviewpoint; this can be regarded as the birth of local class field theory.

1Hasse did not use the word “local”, instead he used “small” (im Kleinen).

61

62 CHAPTER 6. LOCAL THEORY

Chapter 7

Appendix: About Furtwangler

Philip Furtwangler (1869-1940) had an important hand in the creation ofreciprocity and class field theory. But it seems to us that his name is notas widely known as he deserves. Therefore let us first insert some commentsabout Furtwangler’s biography.1

Philipp Furtwangler2 (1869-1940) had studied in Gottingen. In Hofreiter’sbiography [Hof40] we read that Furtwangler had attended, among others, a2-semester course on number theory by Felix Klein. Mimeographed lecturenotes for this course had been prepared by Furtwangler jointly with Som-merfeld. The topic of his thesis “On ternary cubic forms in number theory”had been proposed by Felix Klein. On March 1, 1895 he obtained his Ph.D.3

At that time everybody in Germany who aspired to an academic teachingposition was supposed to do his Habilitation, which is the highest academicqualification and is connected with the right to teach at universities and tosupervise research. But Furtwangler did not. Instead he became assistant

1See the obituaries [Hof40] and [Hub41]. The name of Furtwangler is not yet containedin the Mac Tutor History of Mathematics archive (as of the year 2013). Some personalrecollections on Furtwangler can be found in in the autobiography of Olga Taussky [TT80].Furtwangler had been her academic teacher.

2He was a cousin of the well known conductor Wilhelm Furtwangler.3Olga Taussky in her memories [TT80] says that Furtwangler had no formal training

in number theory. This contradicts what we have read in Hofreiter’s biography [Hof40]. Itappears that she got her information about Furtwangler’s past from hearsay only and notfrom her teacher directly. (When she got her Ph.D. in 1930 she was 24, and Furtwanglerwas 61.)

63

64 CHAPTER 7. APPENDIX: ABOUT FURTWANGLER

at the Physics department of the Technical College in Darmstadt, and laterhe held a position at the Geodesics Institute in Potsdam. He had severalpublications in Geodesics and related fields, among them an article in theEnzyklopadie der mathematischen Wissenschaften.4 In 1904 he accepted aposition as professor of Mathematics at the Agricultural College in Bonn,and later for some years at the Technical University in Aachen. We see thathe was occupied in those years mainly with applied Mathematics. However,as he explained later in a handwritten Curriculum Vitae:5:

Privatim setzte ich meine zahlentheoretischen Studien fort, um einevon der Gottinger Gesellschaft der Wissenschaften gestellte Preisauf-gabe zu bearbeiten. . .

Privately I continued with my studies in number theory, in orderto work on a prize problem announced by the Society for Scienceat Gottingen. . .

We have already reported in the introduction that Furtwangler had beenawarded the prize in 1901 although his solution was not yet complete. Fromhis C.V. we can see that he continued to work on the problem in order toobtain the complete solution without any restrictive hypothesis.

Step by step after several papers, he finally succeeded. His final paperwas split into three parts: [Fur09, Fur12, Fur13], the last one appearing 1913covering the case ` = 2. 6

Without doubt this was a great achievement, in particular if we considerthat Furtwangler could do his number theoretical research in his free timeonly. Moreover he worked quite alone with no one to consult. Consequently

4In [TT80] it is said that Furtwangler “had started off with geodesics during WorldWar I.” Again, we conclude that this information is not quite correct. It may well be thatFurtwangler had returned to geodesics during World War I but the start of his work ingeodesics had been much earlier. By the way, Furtwangler did not stop his research onreciprocity during war time. He reports in [Fur26] that in 1916 already he had succeededto prove the reciprocity law for the exponent m = `2, i.e., the square of a prime number`. He considered this as a prototype of the reciprocity law for arbitrary exponent m.

5Universitats-Archiv Wien, PH PA 1697.6We have said above that the case ` = 2 had been covered by Hilbert. That is only

partially true, for in some details Hilbert had been quite sketchy; therefore Furtwanglerprovided a complete solution also for ` = 2.

65

his progress had been quite slow. In a letter to Hilbert, dated Jan 29, 1902he wrote:

Ich bedauere, dass es mir nicht fruher moglich war, die Arbeit druck-fertig zu machen; doch wenn man taglich 6 Stunden Institutsdiensthat und wenn dann noch sonstige unaufschiebbare Arbeiten dazu-kommen, so hat man kaum Zeit, an andere Dinge zu denken. Ichbedauere dies um so mehr, als ich gerade an zahlentheoretischenUntersuchungen lebhaftes Interesse habe.

I regret that it was not possible for me to complete the paperearlier for printing7; but when there are 6 hours daily of workat the Institute 8, and when in addition there appears even morework which cannot be postponed9 then there is little time to thinkabout other things. I regret this in particular since I am vividlyinterested in number theoretical investigations.

Hilbert’s interest in those years had shifted from number theory to othermathematical fields : foundations of geometry, integral equations and func-tional analysis, later also the theory of variations and mathematical logic.Nevertheless he encouraged Furtwangler in his work on number theory. Inthe Hilbert Nachlass which is preserved in the Handschriftenabteilung of theGottingen University Library I have found 12 letters from Furtwangler toHilbert during the years 1901–1911, and in one of those letters, dated Jan-uary 26, 1903, Hilbert had noted at the end of the letter:

Furtwangler ist der einzige, der sich in die Korpertheorie so hinein-gearbeitet hat, dass er sie selbst zu fordern im Stande ist.

Furtwangler is the only one who has familiarized himself with fieldtheory so much that he himself is capable to advance it.

This sounds like a memo which Hilbert had noted for himself for use in someletter of recommendation for Furtwangler, or similar occasion.10

7That paper appeared 1902 in the Gottinger Abhandlungen [Fur02].8Furtwangler refers to his job at the Institute for Geodesics in Potsdam.9It seems that Furtwangler refers to his engagement in the edition of the “Enzyklopadie

der mathematischen Wissenschaften”, concerning articles on Geodesics and related ques-tions.

10When Hilbert wrote “field theory” then he meant “class field theory” in this context.


In the year 1912 Furtwangler received an offer for a professorship at theUniversity of Vienna. At that time he had already become well known tothe mathematical community in view of his successful work on Hilbert’s9-th problem. In acknowledgement of this important work the University ofVienna had extended its offer to Furtwangler although he had never done hisHabilitation. This is a rare situation in the history of German universities.

It seems that Hilbert had some share in this process. We conclude thisfrom a postcard which Furtwangler sent to Hilbert on Juli 25, 1911. Therehe wrote:

Herzlichen Dank fur Ihren freundlichen Gluckwunsch, der allerdingsnoch etwas zu fruh kommt! Einen offiziellen Ruf habe ich noch nichterhalten . . . Ich wurde mich naturlich sehr freuen, wenn die Verhand-lungen zu einem guten Ende fuhren wurden und ich Algebra & Zah-lentheorie an einer Universitat vertreten konnte.

Many thanks for your kind congralution, which however is a some-what too early! I have not yet received an official offer . . . Ofcourse I would be very glad if the negotiations would come to ahappy ending and I would be able to represent algebra & numbertheory at a university.

We conclude that Hilbert had been informed from Vienna that Furtwanglerindeed will get the offer for the chair and so he had sent him his congratula-tion, although Furtwangler himself had not yet received the offer.

Furtwangler stayed at Vienna University until his retirement in 1938.He was known as a brilliant lecturer; his courses were always crowded withstudents – so much that sometimes tickets for the Furtwangler lecture had tobe issued, as Hofreiter reports [Hof40]. In the 1920s Furtwangler contracteda disease which in the course of time made him heavily paralysed, from theneck downwards. He had to be carried into the lecture hall and lectured fromthe wheelchair; the formulas had to be written to the blackboard by one ofhis assistants. He lectured without manuscript.

Kurt Godel, who was one of his students, said later that Furtwangler’slectures were the best he ever attended. When Artin in his first semester1916/17 studied in Vienna11 he probably attended one of Furtwangler’s lec-

11In the year 1917 Artin was drafted to the army and hence had to interrupt his studies.

67

tures. In any case, Artin kept contact to Furtwangler and the latter sentone of his best Ph.D. students, Otto Schreier, to Artin in Hamburg (this wasin 1924). We shall see later that in 1928 there arose a kind of cooperationbetween Artin, Schreier and Furtwangler which finally led to the proof of theprincipal ideal theorem of class field theory (see section ??).

* * * * *

wird fortgesetzt

After the war he continued his studies in Leipzig with Herglotz.

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