144 Normat 59:3-4, 144–177 (2011)

Galois groups and number theory

Juliusz BrzeziÒski

Mathematical Sciences

University of Gothenburg and Chalmers University of Technology

S–41296 Göteborg, Sweden

jub@chalmers.se

1 Introduction

Galois theory grew up as a theory of solving polynomial equations. Solutions of

polynomial equations, especially equations with integer or rational co e ﬃcients, de-

ﬁne diﬀerent interesting sets of numbers whose investigation has always been of

great interest in mathematics. One purpose of this article is to show how Galois

groups are related to number theory. In fact, Galois groups are deﬁned by the zeros

of polynomial equations and can be used in many diﬀerent ways for solving deep

problems related to numbers. Galois groups are present in many parts of mathe-

matics and even if their role in the original theory of equations belongs to great

achievements of 19th century, their role in mathematics is extremely active and

central. The other purpose of this article is to convince the Reader about impor-

tance of Galois groups in solving many of the old and new problems in number

theory.

The ﬁrst three sections contain a number of introductory notions concerning in-

tegers in algebraic number ﬁelds and Galois groups. Then we explain s ome fun-

damental results of class ﬁeld theory which counts as the biggest achievement of

mathematics in the beginning of 20th century and shows a close relation between

Galois groups and arithmetic of algebraic number ﬁelds in the abelian case (ﬁeld

extensions with abelian Galois groups). Diﬀerent attempts to generalize class ﬁeld

theory to the non-abelian ﬁeld extensions led after a half century to what is known

as the Langlands Program formulated around 1970. The evidence of the conjec-

tures formulated by Robert Langlands was amazingly limited when his program

was created (like a reconstruction of a dinosaur given a few of his bones ). Nev-

ertheless, the foresightedness of Langlands ideas has already led to a long series

of results which strongly support his program. Through possible generalizations

of some central theorems of class ﬁeld theory, like the Kronecker-Weber Theorem,

we touch one of the Langlands conjectures and show how the proof of Fermat’s

Last Theorem given by Andrew Wiles can be considered as a conﬁrmation of the

Langlands Program.

Normat 3-4/2011 Juliusz BrzeziÒski 145

2 Why extend the integers?

Number theory is one the oldest branches of mathematics primarily interested

in the properties of integers and rational numbers, but as a direct consequence

of this, it is also interested in properties of all other numbers and mathematical

object, which app ear in diﬀerent ways when questions concerning the integers and

rational numbers are studied. This is closely related to the fact that when a possible

solution of a problem concerning, for example, the integers is considered, a method

to handle it may be to extend the integers to a bigger set of numbers. Let us look

at some examples.

An old question concerning existence of the Pythagorean triangles, that is, the

triples of positive integers (x, y, z) solving the equation x

+ y

= z

,maybe

transformed to a ques tion about the existence of complex numbers x +yi such that

(x + yi)(x ≠ yi)=z

If in such a triple, the intege rs x, y have only 1 as a common divisor (that is, are

relatively prime) and are of diﬀerent parity (which is the case when all x, y, z have

no a common divisor > 1), then it is possible to check that x+yi, x≠yi considered

as elements of the bigger ring Z[i] of the Gaussian integers a + bi,wherea, b œ Z,

also have only 1 as a common divisor (up to the “sign” which here can be ±1 as

well as ±i). Assuming ”naively” that in Z[i], like in the integers Z, a product of

two relatively prime numbers is a square only if both factors are squares, we get

that it must exist two integers m, n such that

x + yi =(m + ni)

This gives x = m

≠n

,y =2mn, z = m

+ n

. If we assume that m>nand m, n

are relatively prime of diﬀerent parities, we get in fact all Pythagorean triples and

each triple only once. The “only” not motivated assumption is that the Gaussian

integers behave like the integers and have unique factorization in “prime” factors.

We return to the Gaussian primes in a moment, but let us note that we really have

a unique factorization of Gaussian integers as products of the Gaussian primes and

the above method of solving the Pythagorean equation is correct (e ven if it could

be more formalized and in fact, could be replace d by many other arguments leading

to the same answer).

Not so strange that other similar problems were treated in a similar way. If we

want to ﬁnd all positive integer triples such that x

+ y

= z

, then we can also

try to factorize:

≠ y

=(z ≠ y)(z ≠ Áy)(z ≠ Á

y)=x

where Á = e

2ﬁi

≠1+

≠3

(we have Á

=1and Á ”=1). The last factorization can

be considered as a product in the ring Z[Á ] of so called Eisenstein integers. It is

also a ring with unique factorization into primes of this ring. It is well known that

146 Juliusz BrzeziÒski Normat 3-4/2011

there are no positive integers solving this famous Fermat’s cubic equation. A very

beautiful proof of this fact is given in the book by Nagell [N] (Chap. VII, Sec. 64)

using just the arithmetic of the Eisenstein integers.

At early stages, the development of algebraic number theory was highly motivated

by the ideas to solve the general Fermat’s equation x

+ y

= z

for n>2 using

a s imilar technique, that is, factorizing the left hand side as a product of integers

in the ring Z[Á],whereÁ = e

2ﬁi

(we have Á

=1and Á

”=1for 1 Æ m<

n). The method is really fruitful for very small n, and at the beginning of 19th

century some general “proofs” were presented in which the property of the unique

factorization was attributed to the ring Z[Á] for every n (sometimes by well-known

mathematicians). Unfortunately, this is not the case and very soon mathematicians

recognized that it is necessary to use much deeper properties of similar rings trying

to study the Fermat equations or many other similar problems in number theory.

This leads us to the general theory of algebraic numbers and the Galois groups of

the algebraic number ﬁelds.

3 Algebraic numbers and Galois groups

The general situation can be described in the following way. We start with the ring

of intege rs Z and its ﬁeld of quotients Q – the ﬁeld of rational numbers. As in the

case of Pythagorean triples or the solutions of Fermat’s equation, we want to study

bigger sets of numbers like Z[i] or Z[Á]. This means that we move to a number

ﬁeld L ∏ Q, which is deﬁned by solutions of a polynomial equation. In the case of

Pythagorean triples, it is the ﬁeld Q(i) or in general for Fermat’s equations, these

are the ﬁelds Q(Á),whereÁ = e

2ﬁi

, that is, the ﬁelds containing all solutions of

≠ 1=0for n =2, 3,....

In general, algebraic number theory is interested in studying the algebraic num-

bers. Those are solutions of polynomial equations with rational coeﬃcients. All

such solutions form a subﬁeld Q of the complex numbers and algebraic number

ﬁelds are exactly its subﬁelds L whose dimension as vector spaces over the rational

numbers is ﬁnite

In the ﬁeld Q, we are interested in algebraic integers. Those are the numbers

– œ Q such that – satisﬁes an equation with integer coeﬃcients and the highest

coeﬃcient equal to 1. The integers form a subring of Q, which will be denoted by

Z.IfL is a ﬁeld of algebraic numbers, then the integers in it form the ring Z ﬂ L,

which will be denoted by O

Usually, we start with an algebraic number ﬁeld K µ Q instead of Q, and consider

polynomials f(x) with coeﬃcients in K. Using the solutions –

,–

,...,–

of the

equation f(x)=0in Q, we consider the least subﬁeld L = K

of Q containing K

and all these solutions. Thus L = K

= K(–

,...,–

) µ Q (by K(– ) we denote

the least subﬁeld of Q containing both K and – œ Q). It is clear that the same

ﬁeld L may be deﬁned by many diﬀerent polynomials f.

Using the term algebraic number ﬁeld, we shall always mean a number ﬁeld of ﬁnite dimension

over the rational numbers.

Normat 3-4/2011 Juliusz BrzeziÒski 147

The ﬁelds L = K

deﬁned by polynomials f(x) œ K[x] are particulary important

and play very central role in algebraic number theory. They are exactly the (ﬁnite)

Galois extensions of the ﬁeld K. In general, if K is an algebraic number ﬁeld, then

its (ﬁnite) Galois extensions are the ﬁelds L such that L = K

= K(–

,...,–

where –

,...,–

are all solutions of an equation f(x)=0,wheref(x) is a polyno-

mial of degree n with coeﬃcients in K. We say that L is a splitting ﬁeld of f (x)

over K.TheGalois group G(L/K) is the group consisting of all automorphisms of

L over K, that is, bijective functions ‡ : L æ L such that ‡(– + —)=‡(– )+‡(—),

‡(–—)=‡(–)‡(—) for –, — œ L and ‡(a)=a for a œ K. The Galois exten-

sions K ™ L are characterized by the property that the order of the Galois group

G(L/K), that is, the number of automorphisms of L over K equals the dimension

of L as a vector space over K. In the sequel, we will not use this more general

context assuming that K = Q. Thus, a typical situation will be that we look at

ﬁnite ﬁeld extensions K ´ Q.

All automorphisms ‡ :

Q æ Q form the Galois group G(Q/Q) of Q over Q and it

is not diﬃcult to check that if K is a Galois extension of Q, then the restriction

of automorphisms ‡ œ G(Q/Q) to K gives a homomorphism G(Q/Q) æ G(K/Q).

Let’s look at some speciﬁc examples.

Example 1 (a) In case of the Pythagorean triples (the Fermat equation of degree

2), we extended Q by the solutions –

= i, –

= ≠i of f(x)=x

+1 = 0. In this case,

we have K = Q

= Q(i, ≠i)=Q(i). This ﬁeld consists of all numbers a + bi,where

a, b œ Q.TheﬁeldK is of course a Galois extension of the rational numbers and

has degree 2 over Q (quadratic extension). The Galois group G(K/Q) consists of

two elements – the identity and the complex conjugation, that is, ‡(a+bi)=a≠bi.

The integers in the ﬁeld K = Q

are the numbers –, which satisfy equations of

degree at most 2 with integer coeﬃcients and the highest coeﬃcient 1. Thus, if

– = a + bi,wherea, b œ Q is an integer in Q(i), then if it is not simply “the

old” integer a (which satisﬁes the equation x ≠ a =0), it satisﬁes the equation

≠ 2ax + a

+ b

=0with integer coeﬃcients 2a and a

+ b

. It is easy to check

that it implies that a, b must be rational integers. We get that the integers in Q(i)

are exactly the Gaussian integers a + bi,wherea, b œ Z which form the ring Z[i].

(b) It is a little more diﬃcult to ﬁnd the integers in the ﬁeld used for the studies

of the Fermat equation x

+ y

= z

when n>2. As we noted, in this case, we

take f(x)=x

≠ 1=0.IfÁ = Á

= e

2ﬁi

, then all solutions of f(x)=0are Á

for k =1,...,n and K = Q

= Q(Á, Á

,...,Á

)=Q(Á).TheﬁeldK is of course a

Galois extension of the rational numbers, but it is not evident (even if intuitively

clear) that its degree over Q is Ï(n) – the value of the Euler function at n. In fact,

the Galois group consists of the Ï(n) automorphisms ‡

(Á)=Á

,wherek and n

are relatively prime (that is, Á

must be a generator of the cyclic group of order

n generated by Á). Thus the Galois group G(Q(Á)/Q) is the group of residues k

modulo n relatively prime with n, that is, the group (Z/nZ)

. It is a little more

diﬃcult to show that in this case, the integers O

= Z ﬂ K form the ring Z[Á],

that is, every integer in Q(Á) has the form – = a

+ a

Á + ···+ a

n≠2

,where

,...,a

n≠2

œ Z.TheﬁeldsQ(Á),whereÁ = e

2ﬁi

(n>2) play an extreme ly

important role in number theory and are called cyclotomic ﬁelds (over Q).

148 Juliusz BrzeziÒski Normat 3-4/2011

,whenf (x) œ Z[x]

is given. The situation in the examples above shouldn’t mislead. An important

example in this text are quadratic ﬁelds Q(

d),whered is square-free integer. The

integers in such a ﬁeld are a + bÊ,wherea, b œ Z and Ê =

+



,where=d

when d = 1 (m od 4) and =4d when d =2or 3 (mod 4).Thenumber is called

the discriminant of K = Q(

d) (see the general deﬁnition of the discriminant in

section 6). 2

4 Factorization in rings of algebraic integers

When we extend the integers Z to a bigger ring of integers in an algebraic number

ﬁeld K, then the c entral problem is what happens with the prime numbers. We

take a prime number p and consider it as an element in the ring of the integers

. One can ask whether it is possible to factorize p as a product of numbers in

and how such factorizations can be described. It appears that there are some

unexpected diﬃculties, since in general, the unique factorization property of the

integers can not be expected in rings of integers of algebraic number ﬁelds. Before

we look at some examples, let us start with a few general deﬁnitions.

Let K be an algebraic number ﬁeld. In general, we say that a number — œO

divides anumber– œO

if there is “ œO

such that – = —“. The usual

notation is: — | –. A number of O

is a unit if its inverse is also in O

(for

example in Z the only units are ±1). We say that a nonzero number – œO

irreducible if it is not a unit and when – = —“,where—,“ œO

,then— or “

is a unit. We say that – ”=0has a nontrivial factorization if – = —“,where

—,“ œO

and both factors are not units. We can also deﬁne a prime in O

as a

non-unit which divides a product —“ if and only if it divides one of its factors — or

“.Ifﬁ is a prime, then Áﬁ,whereÁ is a unit, is also a prime. Two primes ﬁ and Áﬁ

are called associated. We say that a ring O

has unique factorization if every

nonzero number in O

, which is not a unit, can be written as a product of primes

(in O

) uniquely up to the order of the factors and their associations.

Let us look at some examples. If K = Q(i),thenO

= Z[i] is the ring of Gaussian

integers. It is well-known that this ring has unique factorization. We shall not prove

this fact but we look at some of its manifestations. We see that 2=(1+i)(1 ≠i),

5=(1+2i)(1 ≠ 2i), 13 = (2 + 3i)(2 ≠ 3i) can be factorized. But is not diﬃcult to

prove that 3, 7, 11 can not be represented in a similar way. In fact, if p is a prim e

and p =(x + yi)(z + ti) where x, y, z, t are integers, then taking the square of the

absolute value, we get p

=(x

)(z

). Hence, either x

= z

= p and

then p =(x + yi)(x ≠yi) or one of the factors x

+ y

+ t

is 1 and the other one

. In the last case, one of the numbers x + yi, z + ti is ±1 or ±i. Thus the possible

representations of p as a product of two Gaussian integers, are up to a “sign”, that

is, up to a factor ±1, ±i,eitherp =(x + yi)(x ≠ yi) (nontrivial factorization) or

p = p · 1 (trivial factorization). In the ﬁrst case, we have p = x

+ y

, that is, p

is a sum of two integer squares. Thus p has a nontrivial factorization in Gaussian

integers if and only if p is a sum of two integer squares. But 3, 7, 11 are not sums

Normat 3-4/2011 Juliusz BrzeziÒski 149

of two integer squares, which one checks immediately. Hence 3, 7, 11 can not be

factorized in Z[i] in a nontrivial way.

Remark 1 As we see, the question concerning possibility to factorize prime num-

bers in Z[i] is closely related to an interesting mathematical problem. In fact, this

problem – which prime numbers can be written as sums of two integer squares

– was studied by Fermat who formulated the answer and Euler, who gave a ﬁrst

documented proof of Fermat’s statement (for history of the problem see [C], pp.

8–12). The answer given by Fermat is:

Theorem 1 (Fermat) A prime number p is a sum of two integer squares if and

only if p =2or p = 1 (mod 4).

Fermat’s theorem and the considerations which precede the Remark show the fol-

lowing:

Corollary 1 Aprimep has a nontrivial factorization in Gaussian integers if and

only if p =2or p = 1 (mod 4).

We prove Fermat’s theorem in the Appendix. 2

Let us look at another example, which shows that the problem of fac torization is

somewhat delicate. In the quadratic extension deﬁned by f (x)=x

+5, that is, in

K = Q

= Q(

≠5), the integers are a+b

≠5, a, b œ Z (see the end of section 3). It

is easy to check that the only units of Z(

≠5) (the numbers which have the inverse

in this ring) are ±1 (they are the integer solutions of the equation x

+5y

=1).

If a prime number p is reducible, then p =(x + y

≠5)(z + t

≠5) where x, y, z, t

are integers. Taking squares of the absolute values, we get p

=(x

+5y

)(z

+5t

Hence, either x

+5y

= z

+5t

= p and then p =(x+y

≠5)(x≠y

≠5) = x

+5y

or one of the factors x

+5y

+5t

is 1 and the other one p

. In the last case, one

of the numbers x + y

≠5,z+ t

≠5 must be ±1. Thus the possible factorizations

of p in Z[

≠5], are up to a sign, either p =(x + y

≠5)(x ≠ y

≠5) = x

+5y

(nontrivial factorization) or p = p · 1 (trivial factorization). So for example the

primes 2 and 3 are irreducible, since they c an not be written in the form x

+5y

But this is not “all the truth”. Consider the following equality:

(1) 9=3· 3=(2+

≠5)(2 ≠

≠5).

It is easy to check that all the numbers 3, 2 ±

≠5 are irreducible. B ut they do

not have the most important property of primes, since for example 3 divides the

product (2+

≠5)(2≠

≠5), but of course, neither of its factors. The integer prime

3 remains irreducible, but looses the property which is fundamental in the proof of

the unique factorization of integers into a product of primes. It is not a prime in

the ring Z[

≠5]. Thus this ring does not have unique factorization. The number

9 can be written as a product of irreducible numbers in two essentially diﬀerent

ways.

150 Juliusz BrzeziÒski Normat 3-4/2011

Remark 2 There are many rings of algebraic integers which have unique factor-

ization. It is not diﬃcult to prove that in any ring of algebraic integers O

every

non-unit which is nonzero can be written as a product of irreducible numbers. If

only every irreducible number is prime, then the unique factorization holds. This

is true for the integers in Q(

d) for d = ≠1, ≠2, ≠3, ≠7, ≠11, ≠19, ≠43, ≠67, ≠163

(and for no other negative square-free d) and there is a conjecture of Gauss saying

that it holds for inﬁnitely many positive square-free d, but it is a longstanding

old (since more than 200 years) open question. Many other rings O

have unique

factorization, but the number of know n s uch rings is so far ﬁnite. 2

The lack of unique factorization was a motivation for development of much more

subtle and very deep ideas during the 19th century very often in connection with

diﬀerent attacks on the Fermat’s question concerning his equation. One of the

main contributors was Edward Kummer who solved Fermat’s problem for many

exponents n introducing a new notion of “ideal numbers”, which later in the works

of Richard Dedekind, received the new name “ideals”. They were “ideal” since they

rescued the unique factorization, but in somewhat diﬀerent terms – not the terms

of “speciﬁc numbers” but rather some distinguished sets of numbers.

Dedekind looked at sums of multiples of ﬁxed irreducible numbers. Later the notions

crystalized and an ideal in the ring O

was deﬁned as all sums of multiples of

arbitrary ﬁxed numbers. So if we ﬁx –

,...,–

œO

, then an ideal is the set of

all sums r

–

+ ···+r

–

,wherer

,...,r

œO

. We shall write I = È–

,...,–

Í.

Ideals with one generator, that is, ideals I = È–Í (all multiples of –) are called

principal ideals. Notice that — œÈ–Í means that there is “ œO

such that

— = –“, which means that – divides —, denoted by – | —. Thus the equality

È–Í = È—Í means that – | — and — | –, which is equivalent to — = Á–,whereÁ is a

unit in O

Ideals can be multiplied. If J = È—

,...,—

Í,thenIJ is deﬁned as the ideal which

consists of sums of multiples of all elements of I multiplied by the elements of J.

With the notations above IJ = È–

—

,...,–

—

,...–

—

Í. Among all ideals in O

some play especially important role and replace prime numbers. They are called

prime ideals. An ideal p in O

is called prime if p ”= O

(we say p is proper)

and for any two numbers –, — œO

if –— œ p,then– œ p or — œ p. This is exactly

the divisibility property of primes translated to ideals (if p|ab,thenp|a or p|b)when

the ideal p is principal.

Consider our example in which the unique factorization failed: Z[

≠5]. Take the

two prime ideals (it is not diﬃcult to check that they are prime):

= È3, 1+

≠5Í, p

= È3, 1 ≠

≠5Í.

We have

(2) p

= È9, 3(1 ≠

≠5), 3(1 +

≠5)Í = È3Í

Normat 3-4/2011 Juliusz BrzeziÒski 151

= È9, 3(1 +

≠5), (1 +

≠5)

Í = È1+

≠5ÍÈ1 ≠

≠5, 3, 1+

≠5Í = È1+

≠5Í

(it is easy to check that È1≠

≠5, 3, 1+

≠5Í = È1Í), and similarly, p

= È1≠

≠5Í.

Hence, the left hand side in (1) gives

È9Í = È3ÍÈ3Í = p

· p

= p

while the right hand side

È9Í = È1+

≠5ÍÈ1 ≠

≠5Í = p

Thus even if the number 9 has two diﬀerent presentation as products of irreducible

numbers, the ideal È9Í has exactly one presentation as a product of prime ideals (up

to the order of the factors). The argument above is not a proof, but it is possible

to prove that in all rings of integers O

in the ﬁelds K this is the case (see [C], p.

100):

Theorem 2 Let K be a ﬁeld of algebraic numbers and O

its ring of integers.

Then every proper nonzero ideal I in O

is a product of prime ideals, which is

unique up to the order of the factors:

I = p

···p

where p

are diﬀerent prime ideals in O

Example 2 Let us return to the Gaussian integers Z[i]. This ring is Euclidean

(with respect to the absolute value) and as such has unique factorization. This

implies that each ideal is principal (generated by one number). As we already

noticed, prime numbers behave in three diﬀerent ways in this ring.

If p = 1 (mod 4),thenp =(x+yi)(x≠yi) by Theorem 1. It is not diﬃcult to check

that the ideals p

= Èx + yiÍ and p

= Èx ≠ yiÍ are prime and diﬀerent. Of course,

we have ÈpÍ = p

,soÈpÍ splits into a product of two diﬀerent prime ideals.

If p = 3 (mod 4) and if we have ÈpÍ = Èx + yiÍÈz + tiÍ for some integers x, y, z, t,

then p = Á(x + yi)(z + ti),whereÁ is a unit in Z[i]. As we know, such an equality

implies that either p = x

+ y

= z

+ t

is a sum of two integer squares or some

of x + yi, z + ti is a unit. By Theorem 1, it must be the second possibility, since p

is not a sum of integer squares. This means that the ideal ÈpÍ remains prime – can

not be split into a product of other proper ideals (notice that ÈÁÍ = Z[i] for any

unit Á).

Of course, for the prime 2, we have È2Í = p

,wherep = È1 ≠iÍ,since2=i(1 ≠i)

152 Juliusz BrzeziÒski Normat 3-4/2011

5 An important example

The three possible behaviors of primes in the Gaussian integers are in fact typical

in general situations. But in this section, we look closer at the Gaussian integers,

since they have more interesting properties, which illustrate the general notions.

From the last example, we know that if p = 1 (mod 4),thenÈpÍ = p

is a product

of two diﬀerent prime ideals, while for p = 3 (mod 4) the ideal ÈpÍ remains prime

in Z[i].Suchprimesp are unramiﬁed in Z[i],while< 2 >= p

for a prime ideal

p and in the presence of the e xponent bigger than 1, we say that 2 is ramiﬁed

in Z[i]. Moreover, we say that the primes p = 1 (mod 4) split completely.We

formulate these notions in full generality in the next section.

This behavior of primes p in the Gaussian integers can be explained by the behavior

of the equation x

+1 = 0 (mod p). As we know, the residues Z

= {0, 1, 2,...,p≠

1} of the integers modulo p form a ﬁeld (with addition and multiplication modulo

p). The nonzero residues in Z

form a multiplicative group of order p ≠ 1 and

according to Lagrange’s theorem x

p≠1

=1for each nonzero x (this is simply

Fermat’s Little Theorem). Thus the equation x

p≠1

≠ 1=0has p ≠ 1 diﬀerent

solutions in Z

. In this ﬁeld, we have the factorization:

p≠1

≠ 1=

p≠1

≠ 1

p≠1

which gives that half of the residues are zeros of the second factor. If now p =1

(mod 4) and a is any zero of the second factor, then

p≠1

+1=

p≠1

+1=0,

so x = ±a

p≠1

are (diﬀerent) solutions of the equation x

+1 = 0 modulo p.Observe

that proving this statement, we have used the assumption p = 1 (mod 4) (and this

is the decisive point).

Assume now that p = 3 (mod 4),sayp =4k+3 for an integer k, and x is a solution

of x

+1 = 0 modulo p.Thenx

p≠1

= x

2(2k+1)

=(≠1)

2k+1

= ≠1, which contradicts

Fermat’s Little Theorem. Thus the equation x

+1=0can not be solved modulo

p when p = 3 (mod 4).

If p =2,thenx

+1=(x + 1)

=0of course has a solution x =1(and it has

multiplicity 2).

We can formulate our conclusions in the following way:

Lemma 1 The polynomia l f(x)=x

+1 modulo p is irreducible if p = 3 (mod 4),

a product of two linear factors if p = 1 (mod 4), and a square if p =2.

Thus the factorization of prime numbers p in the Gaussian integers can be perfec tly

explained by diﬀerent patterns of factorizations of the polynomial x

+1modulo

p: The ideal ÈpÍ remains prime if x

+1 is irreducible, is a product of two diﬀerent

Normat 3-4/2011 Juliusz BrzeziÒski 153

prime ideals when x

+1 is a product of two linear factors and is a square of a

prime ideal when x

+1is a square.

If p is an odd prime, then the solvability of the equation x

+1 = 0 (mod p), that is,

= ≠1(modp) is usually expressed in terms of the Legendre symbol.Inthis

case

≠1

. This is deﬁned as 1, w hen the equation is solvable and ≠1 otherwise.

For odd primes p, Lemma 1 says:

≠1

=(≠1)

p≠1

which is a special case of the law of quadratic reciproc ity for the Legendre

symbol. For more about the symbol see the Appendix.

Thus we conclude that the behavior of odd primes in the Gaussian integers depends

on the residues 1, 3 modulo 4. This residues modulo 4 form a group (Z/4Z)

= {1, 3}

– the group of residues modulo 4 relatively prime with 4. There is a nontrivial

homomorphism of this group (in fact injection) ‰

:(Z /4Z )

æ C

,where‰

(1) =

1,‰

(3) = ≠1.ItistheDirichlet character with conductor 4 (see the Remark

on Dirichlet characters at the end of this section). Notice that

‰

(p)=

≠4

≠1

=(≠1)

p≠1

Now we turn to the Galois group G(Q(i)/Q)={1,‡},where‡(z)=¯z is the

complex conjugation. The question is: Is there a re lation between diﬀerent prime

numbers and the elements of the Galois group as regards the behavior with respect

to factorization? If p is a prime having a factorization p =(x + yi)(x ≠ yi),then

the conjugation shifts the factors if only p is odd. If p remains prime then both

automorphisms in the Galois group ﬁx the factors of the trivial factorization p =

p · 1. But we already know that we should look at factorizations of ideals rather

than on factorizations of primes.

If we now look at the action of the automorphism on the ideals rather than on the

factors, we see that if p = 1 (mod 4), then only the identity in the Galois group

maps the ideals p

= Èx + yiÍ and p

= Èx ≠ yiÍ (factors of ÈpÍ) onto itself (the

conjugation shifts them). If p = 3 (mod 4), then both automorphism in the Galois

group map the prime factors of ÈpÍ (there is only one – ÈpÍ) onto itself.

Thus we attach the automorphism 1 to the primes p = 1 (mod 4), and the auto-

morphism ‡ to p = 3 (mod 4). The attached element of the Galois group (in fact,

the generator of the group which ﬁxes the factors of the factorizations of ÈpÍ)is

called the Frobenius automorphism corresponding to p and denoted by Fr

Thus Fr

=1if p = 1 (mod 4), and Fr

= ‡ if p = 3 (mod 4) (we assume that

p ”=2). We shall see in a moment that the deﬁnition of Fr

is very natural and still

better motivated but the case of Gaussian integers is too simple to see the whole

picture.

154 Juliusz BrzeziÒski Normat 3-4/2011

Finally we come to a relation between the two aspects of the Fermat’s question.

The Galois group G(Q(i)/Q)={1,‡} has one nontrivial repres entation – the

only nontrivial homomorphism ﬂ : G(Q(i)/Q) æ C

. It is given by ﬂ(1) = 1 and

ﬂ(‡)=≠1. Hence the two observations concerning the behavior of diﬀerent primes

can be expressed in the following way:

(3) ﬂ(Fr

)=‰

(p)

for every odd prime p. In order to gather the information for all primes p, we can

deﬁne suitable “generating functions” – one deﬁned by the representation ﬂ called

Artin L-function, and the other deﬁned by the character ‰

called Dirichlet

L-function:

L(ﬂ, s)=

(1 ≠ ﬂ(Fr

≠s

)

≠1

and

L(‰

,s)=

(1 ≠ ‰

(p)p

≠s

)

≠1

The equalities (3) translate now to the equality:

L(ﬂ, s)=L(‰

,s).

But the result about the behavior of primes manifests a property of all abelian

Galois extensions of rational numbers and, as we shall see, at least conjecturally,

all ﬁnite Galois extensions of algebraic number ﬁelds (see sections 9 and 10).

Remark 3 We shall meet several times homomorphisms ‰ :(Z/nZ)

æ C

like ‰

in the text above. Such homomorphisms are called Dirichlet characters (see [IK],

§3.2). A character ‰ is called primitive if it not possible to ﬁnd a proper divisor d |

n such that ‰ is a composition of the natural homomorphism (Z/nZ)

æ (Z/dZ)

and a character Â :(Z/dZ)

æ C

.If‰ is primitive, then n is called its conductor.

Thus 4 is c onductor for ‰

above, since this character can not be obtained as a

composition of the natural homomorphism (Z/4Z)

æ (Z/2Z)

= {1} and the

(only trivial) mapping of the last group in C

. 2

6 What happens with prime numbers in Galois extensions?

The observations from the example presented in the previous section are in fact

rather typical as regards behavior of prime numbers in the rings of integers of

Galois extensions. The general result is as follows (see [C], p. 101):

Normat 3-4/2011 Juliusz BrzeziÒski 155

Theorem 3 If K is a Galois extension of Q of degree n, then for any prime number

p, we have

=(p

···p

)

where p

are diﬀerent prime ideals in O

and ref = n, where f is the degree of

the ﬁeld O

/p over Z/pZ and p is any of the ideals p

We say that p is ramiﬁed if e>1, and unramiﬁed if e =1.Itissplit completely

if e =1and r = n ,wheren =[K : Q]. The number of ramiﬁed primes is always

ﬁnite. This terminology applies also to arbitrary algebraic extensions K but then

as we know

= p

···p

and the exponents e

may be diﬀerent. We say that p

is ramiﬁed or unramiﬁed

depending on e

> 1 or e

=1. Aprimep is completely split if all e

=1and r = n.

There is only a ﬁnite number of primes p for which it may happen that e

> 1 (see

the Remark at the end of this section).

How to ﬁnd factorization of a prime number in a Galois extension K of the rational

numbers? There are several results, which give computationally practical methods.

One of them, we use in the examples below (see [C], p. 100, [Ko], p. 81):

Theorem 4 Let K = Q(–) be a Galois ﬁeld over the rational numbers and assume

that – is an integer in K having the minimal polynomial f(x) œ Z[x].Ifp is a

prime number such that the polynomial f(x) modulo p has all diﬀerent zeros (as

a polynomial over the ﬁnite ﬁeld Z/pZ ), then p is unramiﬁed in K. Moreover, if

f(x)=f

(x) ···f

(x) is a product of irreducible polynomials in (Z/pZ)[x], then all

have the same degree and

= p

···p

where p

are prime ideals, p

= Èp, f

(–)Í in O

.Thusp splits completely in K if

and only if f(x) has a factor of degree 1 modulo p, t hat is, the equation f(x)=0

modulo p has a solution in O

Example 3 (a) Let K = Q(

≠5). Take – =

≠5 with minimal polynomial

f(x)=x

+5.WehaveO

= Z[

≠5]. As we already noticed above, the prime

3 splits (see (2)), and now we easily check that x

+5=(x ≠ 1)(x + 1) modulo

3. According to the theorem above (and as we noted before) È3Í = p

,where

= È3, 1+

≠5Í and p

= È3, 1 ≠

≠5Í. The prime 11 remains prime (the

ideal È11Í is prime in Z[

≠5])sincex

+5 mo dulo 11 can not be factored. We have

È5Í = p

,wherep = È

5Í,sincex

+5 = x

modulo 5. We also have x

+5 = (x+1)

modulo 2, so È2Í = q

,whereq =(2, 1+

≠5). Thus the primes 2 and 5 are ramiﬁed

in Z[

≠5]. These are the only ramiﬁed primes, which is easy to check. The ideal q

156 Juliusz BrzeziÒski Normat 3-4/2011

is not principal. In fact, if q =(2, 1+

≠5) = (a + b

≠5),thena + b

≠5 | 2 and

a + b

≠5 | 1+

≠5, which gives a

+5b

| 4 and a

+5b

| 6, that is, a

+5b

or 2. Both possibilities are easy to exclude.

(b) Let K = Q(Á),whereÁ = e

2ﬁi

for a prime p>2. The minimal polynomial of Á

is f(x)=(x

≠1)/(x≠1) = x

p≠1

+···+x +1.WehaveO

= Z[Á]. It is p os sible to

show that the only ramiﬁed prime is p and ÈpÍ = p

p≠1

,wherep =(p, Á≠1) = (Á≠1)

(this follows easily from the factorization of f (x) and the equality f (x)=(x≠1)

p≠1

modulo p). 2

Remark 4 If K = Q(–) for an algebraic integer –, then in order to ﬁnd all but

ﬁnitely many primes p, which are unramiﬁed in K, it is possible to look at the

discriminant of f (x),whichis

(f)=

1Æi<jÆn

(–

≠ –

)

where –

for i =1,...,n are all zeros of f(x) in C. The discriminant (f) can

be expressed as a polynomial with integer coeﬃcients of the coeﬃcients of f(x).

It is not diﬃcult to prove that if p - (f),thenp is unramiﬁed. This implies, of

course, that the number of ramiﬁed primes in any algebraic number ﬁeld K is ﬁnite.

Notice however that it is possible that some primes dividing (f) are unramiﬁed

anyway. There are several variants of Theorem 4, which can be used for practical

computations (see e.g. [C], p. 102). 2

7 Class ﬁeld theory and its generalizations

As we have seen, a prime p is c ompletely split in Z[i] if and only if p = 1 (mod 4).

This property of primes in the Gaussian integers is, in fact, speciﬁc for the ﬁeld

Q(i) characterizing it among all abelian Galois extension of the rational numbers.

Since description and classiﬁcation of all Galois extensions of the rational numbers

in terms of some kind of arithmetic properties was, and still is, an important task

of algebraic number theory, one can wonder if it is possible to characterize diﬀerent

Galois extensions in terms of similar properties.

Let K be a Galois ﬁeld over the rational numbers. Denote by S

(K/Q) the set of

prime numbers for which the ideal pO

splits completely in O

.IftheﬁeldK is

real, we shall extend the set S

(K/Q) with an extra element Œ called the inﬁnite

prime. The meaning of this deﬁnition will be clear in a moment. The fundamental

observation, which is not very diﬃcult to prove, but which shows the importance

of the sets S

(K/Q) is the following (see [C], Theorem 8.19):

Theorem 5 Let K

and K

be two Galois extensions of the rational numbers.

Then K

´ K

if and only if S

/Q) ™S

/Q). In particular, we have

= K

if and only if S

/Q)=S

/Q).

Normat 3-4/2011 Juliusz BrzeziÒski 157

Thus if we succeed with a some reasonably simple characterization of primes in

the sets S

(K/Q), then we can get a classiﬁcation of Galois extensions of Q.In

fact, diﬀerent attempts to ﬁnd such a classiﬁcation led to developments in algebraic

number theory which culminated at the beginning of 20th century with class ﬁeld

theory. This theory gives an explicit description of the sets S

(K/Q) when the

Galois groups G(K/Q) are abelian. It is still an open and very central problem in

number theory to ﬁnd corresponding results for non-abelian Galois extensions of

algebraic number ﬁelds.

First we give some easy to describe sets S

(K/Q):

Example 4 (a) Let K = Q(i). As we saw before, we have p œS

(K/Q) if and

only if p = 1 (mod 4). As we also know, the set S

(K/Q) gives exactly those odd

primes for which the polynomial f(x)=x

+1is a product of two diﬀerent linear

factors. At the same time, these are those primes which are the sums of two integer

squares.

(b) Similarly as in (a), it is possible to prove that if K = Q(

≠2),thenp œ

(K/Q) if and only if p =1or 3 (mod 8). This happ ens if and only if f (x)=

+2 splits in a product of two diﬀerent linear factors. At the same time, we have

p œS

(K/Q) if and only if p = x

+2y

for some integers x, y.

2ﬁi

,thenp œS

(K/Q) if and only if p =1 (modn). 2

The general pattern seems to be that the primes belonging to S

(K/Q) are char-

acterized by some arithmetical progressions deﬁned modulo an integer in some way

related to the extension Q µ K. In fact, the class ﬁeld theory gives an explana-

tion of this pattern (see Theorem 8). This impressive theory was developed by

the eﬀorts of many distinguished mathematicians. Starting with the works of Fer-

mat, Euler, Lagrange and Gauss, the theory developed during the 19th century in

the works of Kummer, Dedekind, Kronecker, Weber and culminated in the ﬁrst few

decades of 20th century when Hilbert, Takagi and Artin formulated the ﬁnal results

for abelian Galois extensions of algebraic number ﬁelds. During the following years

the class ﬁeld theory was developed in many diﬀerent ways and went through many

diﬀerent contextual transformations depending on the developments in many other

ﬁelds of mathematics. But still a suitable context of possible generalizations in the

case of non-abelian Galois extensions was missing. About 1967, Robert Langlands

formulated several conjectures concerning possible generalizations of the class ﬁeld

theory to arbitrary Galois extensions. In fact, the context of these generalizations

covers much more than Galois groups of algebraic number ﬁelds. We shall not try

to explain this broader perspective (see for example [G]), but only note that to-

day a few of the special cases of these conjectures are proved. We discuss some of

these developments a little closer in the following sections. Now we only mention

two aspects of the abelian case, which are starting points for diﬀerent kinds of

generalizations.

The results for abelian Galois extensions of the rational numbers which give a com-

plete description of the sets S

(K/Q) follow from a theorem which was formulated

by Kronecker in 1853, by Weber in 1886 and ﬁnally proved by Hilbert in 1896. It

says that every abelian extension of the rational numbers is a subﬁeld of a suitable

158 Juliusz BrzeziÒski Normat 3-4/2011

cyclotomic ﬁeld (see section 9 for exact formulations). This result is the ﬁrst which

we want to discuss c lose r in the following sections. It became already famous in

1900 when David Hilbert asked about possible generalizations of it in the twelfth

problem on his list of the 23 mathematical problems for the 20th century.

The second and the most central asp ec t of the theory of algebraic number ﬁelds is

related to the last observations in the example discussed in sec tion 5 – a relation

between the Artin L-function of the Gaussian integers and the Dirichlet L-function

corresponding to a suitable Dirichlet character on the rational integers.

In fact, s uch a relation between Artin L-functions in the case of abelian extensions

of rational numbers (we deﬁne the Artin L-functions below) and the Dirichlet L-

functions deﬁned by characters on the rational integers exists in general. This is

in principle only a diﬀerent formulation of the Kronecker-Weber Theorem. This

theorem, when formulated in terms of L-functions gives also a natural context in

which it is possible to study diﬀerent reciprocity laws known in number theory

beginning with the famous law of quadratic reciprocity. We shall explain these

points in the following sections.

The Artin L-functions were deﬁned in 1925 at the ﬁnal stages of development

of class ﬁeld theory in the abelian case. They are however deﬁned for arbitrary

Galois extensions of number ﬁelds. Moreover, they are not only deﬁned for ho-

momorphisms ﬂ : G(L/K) æ C

, which give representations of the Galois group

G(L/K) of dimension 1, but for arbitrary representations ﬂ : G(L/K) æ GL

(C)

of arbitrary dimension n. If the Galois group is abelian, then all irreducible repre-

sentations have dimension one and there is no need of n>1. But for non-abelian

Galois groups, it is necessary to consider representations of higher dimensions.

Already at the beginning, the theory was extended to arbitrary ground ﬁelds instead

of the rational numbers so that arbitrary abelian Galois extensions of arbitrary

algebraic number ﬁelds were s tudied. The fundamental results of class ﬁeld theory

were formulated in this context, but it was necessary to use a broader class of L-

functions generalizing those corresponding to the Dirichlet characters. These new

L-functions were deﬁned by Hecke using so called Hecke characters in the case of

arbitrary algebraic number ﬁelds as a generalization of Dirichlet characters in the

case of the rational numbers (see [IK], p. 56, [Ko], p. 204).

Still the theory existed only in the case of abelian Galois extensions and the corre-

sponding Artin L-functions. For non-abelian Galois extension it was not clear how

to replace the Hecke (or Dirichlet) characters by other functions deﬁned on some

kind of objects related to the rational numbers (or in general case, to the ﬁelds of

algebraic numbers). These constructions had to await for the Langlands Program

even if some functions, which much later we re incorporated in the non-ab e lian case

were already studied by Hecke approximately at the same time when Artin deﬁned

his functions (at the same university in Hamburg about 1925). But the insight

that the whole theory could be formulated in terms of group representations and

the c orresponding L-functions came much later when Langlands formulated his

program.

We shall try to explain some of the relations in the following sections, but earlier we

have to explain a few facts about Artin maps (Artin symbol, Frobenius element).

Normat 3-4/2011 Juliusz BrzeziÒski 159

8 Frobenius Element and the Artin symbol

Assume that K is a Galois extension of the rational numbers. Denote by S(K/Q)

the set of all primes p, which are unramiﬁed in K, that is,

= p

···p

where r

=[K : Q] for p œS(K/Q).SinceZ modulo p is a ﬁeld, the quotient of

modulo p,wherep is one of the ideals p

, is a ﬁeld e xtension of it (of degree f

Every automorphism ‡ œ G(K/Q), which maps p on p gives an automorphism of

the ﬁeld O

/p and, in fact, we get all automorphisms of this ﬁeld. Moreover, if p is

unramiﬁed, there is only one automorphism of K such that ‡(p)=p, which gives

the given automorphism of O

/p. Now the ﬁeld O

/p, as a ﬁeld of characteristic

p, has an automorphism x ‘æ x

called the Frobenius automorphism of O

/p.

The automorphism of K which deﬁnes the Frobenius automorphism of O

/p is

called the Frobenius element corresponding to p (see [C], p. 106):

Theorem 6 Let Q µ K be a Galois extension, let p be a p rime number unramiﬁed

in K and let p be a prime ideal in the splitting of pO

as a product of prime ideals.

Then there exists exactly one element ‡ œ G(K/Q) such that ‡(x)=x

modulo p

for every x œO

The automorphism ‡ corresponding to p is called the Frobenius element and is

denoted by Fr

K/Q

. But if the Galois group G(K/Q) is abelian, then the

Frobenius eleme nt is in fact the same for all prime ideals appearing in the splitting

of pO

. Therefore, in the abelian case, the Frobenius element is denoted by Fr

K/Q

. It corresponds to the prime number p.

Denoting by I

the multiplicative group of all products p

···p

,wherep

are

primes in the set S(K/Q) and k

are integers, we get a group homomorphism

K/Q

: I

æ G(K/Q)

such that

K/Q

when a = p

···p

. This mapping is called the Artin map and (

K/Q

) is called

the Artin symbol (see [C], p. 106). We have very modest ambitions and restrict

the deﬁnitions to the rational numbers as a ground ﬁeld, but the Artin map is

deﬁned similarly in general case for any algebraic number ﬁeld as the ground ﬁeld.

The Artin map is surjective and the description of its kernel (Artin kernel)plays

160 Juliusz BrzeziÒski Normat 3-4/2011

a very important role in class ﬁeld theory. Since we only need the deﬁnition of the

Frobenius element, we do not need to go deeper into these matters, but several

examples in the next section will give a better insight into these notions.

9 Two versions of the Kronecker-Weber Theorem

Let K = Q(–) be an abelian Galois extension of the rational numbers, where – is

an algebraic integer with minimal polynomial f(x). The prime numbers unramiﬁed

in K play two diﬀerent roles in their relations with the Galois group of K.First

of all, such a prime number p deﬁnes its Frobenius element. Intuitively it is closely

related to the type of the factorization of f (x) modulo prime numbers p.Thisis

not quite correct intuition, since primes with diﬀerent Frobenius automorphisms

can give the same type of the factorization of f(x) (see the last Example in this

section). Nevertheless, for almost all prime numbers p (all with possibly a ﬁnite

number of exceptions), the Frobenius Fr

=1if and only if f(x) modulo p is the

product of linear factors (that is, the ideal pO

splits completely in O

The other role played by primes comes from the Kronecker-Weber Theorem:

Theorem 7 Kronecker-Weber Theorem. Let K be an abelian Galois number

ﬁeld over Q. Then there exists a cyclotomic ﬁeld Q(Á

), which contains K (Á

2ﬁi

The least n such that K ™ Q(Á

) is called the conductor of K and sometimes is

denoted by f

Example 5 The ﬁeld K = Q(

5) has discriminant =5(see the end of section

3). Hence the ﬁeld K is contained in the cyclotomic ﬁeld Q(Á

). In fact, if – = Á

, then we check that f(–)=0,wheref(x)=x

≠x ≠1.Wehave– =

(1 +

5),

so – generates K. In general, the quadratic ﬁeld K = Q(

d) with the discriminant

 (see the end of section 3) is contained in the cyclotomic ﬁeld Q(Á

||

) (see [C],

§8). 2

Let us express this in form of a theorem, which answers one of the most funda-

mental questions concerning the primes in abelian Galois extensions of the rational

numbers:

Theorem 8 Let K be an abelian Galois extension of the rational numbers. Then

there exists n and the residues r

,...,r

relatively prime to n such that a prime

number p is completely split in K if and only if p = r

(mod n) for i =1,...,k.

Proof. According to the Kronecker-Weber Theorem, there exists n such that

K ™ Q(Á

). As we already know, the Galois group G(Q(Á

)/Q) is isomorphic

to (Z/nZ)

. Thus the group G(K/Q) can be represented as a quotient of (Z/nZ)

Normat 3-4/2011 Juliusz BrzeziÒski 161

since G(K/Q)=G(Q(Á

)/Q)/G(Q(Á

)/K). Denote H = G(Q(Á

)/K).Thusevery

prime number p not dividing n deﬁnes an element in the quotient (Z/nZ)

/H.This

element equals 1 if and only if the residue of the prime p in (Z/nZ)

belongs to H.

Moreover, it is not diﬃcult to check that the Frobenius element Fr

corresponding

to a prime number unramiﬁed in K is in the group G(K/Q) equal to the restric-

tion of the Frobenius element Fr

corresponding to the same prime number in the

Galois group G(Q(Á

)/Q). Hence Fr

=1in G(K/Q) if and only if the Frobenius

automorphism ‡

: Á

‘æ Á

(corresponding to the residue of p in the Galois group

G(Q(Á

)/Q)=(Z /nZ )

) re stricts to the identity on K. This happens exactly when

the residue of p is in H. Now the elements of H are some residues r

,...,r

modulo

n relatively prime to n,sop œ H means p = r

(mod n) for i =1,...,k. 2

Example 6 Let K = Q(

≠5). We want to characterize the primes which com-

pletely split in the ring O

= Z[

≠5]. The discriminant of K equals 20 and

according to the Kronecker-Weber Theorem the ﬁeld K is subﬁeld of the cy-

clotomic ﬁeld L = Q(Á

) (see the previous example). Hence, the Galois group

G(L/Q)=(Z/20Z)

= {1, 3, 7, 9, 11, 13, 17, 19}. The subgroup of G(L/Q) deﬁn-

ing the ﬁeld K (as the ﬁeld of ﬁxed elements with respect to the automorphisms

‘æ Á

)isH = {1, 3, 7, 9}. It needs some computations to check that the au-

tomorphisms corresponding to thes e values of r ﬁx

≠5. Hence, the primes which

completely split in K are p =1, 3, 7, 9 (mod 20). An interesting point is that this re-

sult has a relation to the problem of representations of primes by the form x

+5y

The number 5 is here an exception (ramiﬁed in K) and those represented are ex-

actly p =1, 9 (mod 20), which was a conjecture formulated by Euler. The primes

p =3, 7 (mod 20) are represented by the form 2x

+2xy +3y

. This has a very in-

teresting explanation and is related to so called Gauss genera – a theory developed

by Gauss in his studies of binary quadratic forms (see [C], p. 33). 2

As we already saw in section 5, the two ways in which prime numbers deﬁne the

elements of the Galois group (Frobenius or residue class), result in a relation be-

tween two types of L-functions (the Artin L-function and the Dirichlet L-function).

This is a general phenomena which can be encoded directly into another very fre-

quent formulation of the Kronecker-Weber Theorem, which opens directly a way

to natural questions giving ve ry fruitful generalizations:

Theorem 9 Kronecker-Weber Theorem. Let K be an abelian Galois number

ﬁeld over Q and let ﬂ : G(K/Q) æ C

be any homomorphism. There exists a unique

primitive Dirichlet character ‰

ﬂ

:(Z /f

ﬂ

æ C

such that for each prime number

p relatively prime to f

ﬂ

, we have

(4) ﬂ(Fr

)=‰

ﬂ

(p).

As we saw in se ction 5, in order to gather the information for all primes p,we

can deﬁne suitable “generating functions” – one deﬁned by the representation ﬂ

called Artin L-function, and the other deﬁned by the character ‰

ﬂ

called Dirichlet

L-function:

162 Juliusz BrzeziÒski Normat 3-4/2011

L(ﬂ, s)=

(1 ≠ ﬂ(Fr

≠s

)

≠1

and

L(‰

ﬂ

,s)=

(1 ≠ ‰

ﬂ

(p)p

≠s

)

≠1

The equalities (4) translate now to the equality:

L(ﬂ, s)=L(‰

ﬂ

,s).

This last version is essentially equivalent with the ﬁrst formulation of the theorem

(see [BG], p. 13, Theorem 5.2 and Corollary 5.3). Moreover, its particular cases

correspond to the reciprocity laws in diﬀerent versions. We look at the best known

case of the law of quadratic reciprocity (for the Legendre symbol) in the Appendix.

The last result is also a starting point of many far going generalizations. On the “left

hand side” we can replace the representation ﬂ by any (irreducible) representation

G(L/Q) æ GL

replaced by any algebraic number ﬁeld. On the “right hand side”, the Dirichlet

characters can be replaced by suitable other types of functions corresponding to

the representations chosen on the “left hand side”. We develop this ideas a little

more in the following sections.

Example 7 Let f(x)=x

+ x

≠ 2x ≠ 1 and K = Q(–),wheref(–)=0.Then

the solutions of f(x)=0are f

(–)=–, f

(–)=–

≠ 2 and f

(–)=1≠ – ≠ –

Moreover, the discriminant (f)=7

and the ring of integers of K is O

Z ﬂ K = Z[–].TheﬁeldK is contained in the cyclotomic ﬁeld L = Q(Á

). In fact,

the example was constructed using this ﬁeld of degree 6 over the rational numbers

and computing the subﬁeld ﬁxed by the automorphism Á

‘æ Á

=1/Á

=¯Á

order 2, which is the complex conjugation. Thus the ﬁxed ﬁeld K has degree 3 over

the rational numbers.

The Galois group G(L/Q)=(Z/7Z)

= {1, 2, 3, 4, 5, 6}. The subgroup deﬁning K

consists of the automorphisms corresponding to 1 and 6 (the identity and Á

‘æ Á

Thus a prime p is completely split in K if and only if p =1, 6 (mod 7). For example,

we check that

f(x)=(x + 3)( x + 5)(x + 6) (m od 13).

The corresponding Frobenius automorphism is of course the identity – ‘æ – in the

Galois group G(K/Q). For the remaining primes p with the exception of p =7,

which is ramiﬁed (f(x)=(x + 5)

(mod 7)), the polynomial f(x) is irreducible

modulo p. The Frobenius automorphism corresponding to the primes p =2, 5

(mod 7) is – ‘æ –

≠ 2, and corresponding to the primes p =3, 4 (mod 7) is

Normat 3-4/2011 Juliusz BrzeziÒski 163

– ‘æ 1 ≠– ≠–

. This is easy to check computing e.g. –

(mod 7) and –

(mod 7).

10 How to generalize the Kronecker-Weber Theorem?

At the beginning of 20th century the Kronecker-Weber Theorem was already proved

and a natural question was whether it is possible to ﬁnd other “standard” exten-

sions of the rational numbers, which describe, maybe s ome classes, of non-abelian

extensions of rational numbers in a similar way as the cyclotomic ﬁelds describe

all abelian extensions. Another problem known as “Kronecker’s Jugendtraum”

( “youthful dream”) was to ﬁnd ﬁelds that correspond to cyclotomic when the

ground ﬁeld is not the rational numbers, but some extension of it (for example, the

quadratic extensions of the rational numbers). This last problem became especially

famous as twelfth problem, of the 23 mathematical problems announced by David

Hilbert in 1900.

The cyclotomic ﬁelds Q(Á

) are generated by the values Á

= e

2ﬁi

of the exponential

function e

. The purpose was to ﬁnd a function (instead of e

) whose values should

generate a family of ﬁeld extensions over, say, quadratic ﬁelds, which play a similar

role for abelian extensions of these ﬁelds as the cyclotomic ﬁelds play for abelian

extensions of the rational numbers.

The best known case in which this problem was solved is the case of quadratic

imaginary ﬁelds (non-real quadratic extensions of the rational numbers) in which

case Kronecker’s Jugendtraum has been solved by the theory of “complex multi-

plication” (see [C], §14 and [S]). One of the functions whose values are needed to

generate abelian extensions is the j-function (best known from the theory of elliptic

curves) but the complete answer needs some other functions as well (see [S]). Even

if there are partial results in particular cases, notably for so called CM -ﬁelds or

real quadratic ﬁelds, the problem is largely still open.

But the second formulation of the Kronecker-Weber Theorem (Theorem 9) opens

also for generalizations, which are very fruitful. At the beginning of 20th ce ntury,

it was still unclear how to generalize the statement to arbitrary Galois extensions

of the rational numbers or even abelian e xtensions when the ground ﬁeld is an

arbitrary algebraic number ﬁeld. Therefore the problem could not ﬁnd a place on

the list of Hilbert problems. But during the ﬁrst few decades of 20th century the

main results of the theory of abelian Galois extensions over arbitrary algebraic

number ﬁelds were accomplished as we related in section 7.

Let us recall that in the classical case of the class ﬁeld theory of abelian extensions

K, the representations of the Galois group G(K/Q) have dimension 1, that is,

they are homomorphisms in the group C

. In general case, when the Galois group

is not nece ss arily abelian, there are linear representations of higher dimensions

ﬂ : G(K/Q) æ GL

representation of the Galois group G(K/Q), Emil Artin attached a function, which

now is called Artin L-function:

164 Juliusz BrzeziÒski Normat 3-4/2011

L(ﬂ, s)=

det

≠ ﬂ(Fr

≠s

≠1

As a function of a complex variable s, it is nowhere vanishing analytic when Ÿ(s) >

1 (see [BG], Chapter 4). But the expectations go much farther. Notice also that

for simplicity, we consider here only primes p such that p is unramiﬁed in K (if

p is ramiﬁed, then the deﬁnition of the corresponding factor must be modiﬁed).

Moreover, when ﬂ has dimension > 1,thenFr

depends in general on the ideals p in

the splitting of the prime number p. However, the Frobenius elements corresponding

to diﬀerent p are conjugated inside of the Galois group, so the determinants det(I

≠

ﬂ(Fr

≠s

) only depend on p. Many very central problems in number theory are

related to the properties of the Artin L-functions. It is known that L(ﬂ, s) has a

meromorphic continuation to the whole complex plane, but much more is expected

(see [BG], p. 83):

Conjecture 1 (Artin Conjecture) Let ﬂ be an irreducible and nontrivial Galois

representation of G(K/Q). Then L(ﬂ, s) is an entire function in the whole complex

plane.

When the dimension of ﬂ is one and the ground ﬁeld are rational numbers, Artin

proved that his L-function is equal to a suitable Dirichlet L-series (the Galois

ﬁeld K need not be abelian) and in this case his conjecture is true (thanks to

the connections with the Dirichlet and Hecke L-functions). What happens when

the dimension of the representation is greater than 1? Notice that the question

is meaningful for arbitrary Galois extensions K of the rational numbers and, as

we noted before, the rational numbers can be replaced by any number ﬁeld. Is it

possible to ﬁnd a kind of L-functions deﬁned by objects over the integers Z (like

Dirichlet characters of (Z/f

) which are equal to Artin’s L-function (which are

deﬁned in terms of the ﬁeld K)?

As we saw, a possibility to establish an equality between the Artin function for one

dimensional representation ﬂ of G(K/Q) and an L-function of a suitable Dirichlet

character is an equivalent way to express very fundamental results in number theory

sometimes formulated in diﬀerent terms. Thus, one should expect that also in

general, a possibility to establish an equality of the shape:

(5) L(ﬂ, s)=L(‰

ﬂ

,s),

where on the left, we have Artin’s L-function and on the right some hypothetical

L-functions related to a kind of arithmetical objects over the rational numbers (or

over another ﬁeld chosen as a ground ﬁeld) should give deep insights into interesting

problems in number theory. This is really the case even if the formulation is far

from being evident.

The next case could be the Galois representations of dimension 2 and the corre-

sponding L-functions. But the question concerning the corresponding L-functions

Normat 3-4/2011 Juliusz BrzeziÒski 165

and objects related to the rational numbers (or another ground ﬁeld) which should

be chosen on the right hand side of the equality (5) was not clear and Artin could

not ﬁnd a candidate. The insight that a generalization of class ﬁeld theory to the

non-abelian case can be formulated in terms of group representations and corre-

sponding L-functions on the right hand side of (5) came much later and the suitable

notions and their context were created by Robert Langlands in 1967. Langlands

formulated several conjectures, which need several much more advanced theories

and notions than those discussed in this article. But we formulate the ﬁrst of them

and give an example in one special case. Still this case is of great interest and great

importance. Let us formulate what is sometimes called Langlands’ ﬁrst conjec ture

(see [G], p. 203) using some notions, which are not formally deﬁned in this article

(but which we try to explain a little closer in a special case considered in the next

section):

Langlands Conjecture for GL

. Let K be a ﬁnite Galois extension of rational

numbers and ﬂ : G(K/Q) æ GL

group. Then there exists an automorphic cuspidal representation ‰

ﬂ

of the group

over Q such that

(6) L(ﬂ, s)=L(‰

ﬂ

,s).

Notice that the main results of the class ﬁeld theory (as for example formulated in

the Theorem 9 for the case of rational numbers) correspond to the case n =1of

this conjecture. Many mathematicians have worked on the cas e n =2, which is not

solved completely. But studies of this case gave solutions of many interesting and

important problems in number theory.

We do not explain the term “automorphic cuspidal representation”, but we give the

deﬁnitions of these objects in the case, which we discuss in the next section about

the famous result of Wiles, which is related to n =2in the Langlands Conjecture

and resulted in a proof of Fermat’s Last Theorem.

11 Wiles’ Theorem

The famous result of Wiles, which gives a proof of Fermat’s Last Theorem, is

related to the Langlands Conjecture for GL

. As a matter of fact, the main re-

sult proved by Wiles with the assistance of Richard Taylor was a special case of

the Taniyama-Shimura-Weil Conjecture on elliptic curves. The relevant rep-

resentations G(Q/Q) æ GL

L-functions on the left hand side in the equation (5) related to Wiles’ work are

deﬁned by some elliptic curves E. On the right hand side of the equation (5) are

L-functions corresponding to so called modular cusp forms of weight 2 for a dis-

crete subgroup 

(N) of GL

(R),whereN is the conductor of the elliptic curve

E. In the next section, we explain how the cases n =1(class ﬁeld theory) and

166 Juliusz BrzeziÒski Normat 3-4/2011

n =2(e.g. Taniyama-Shimura-Weil Conjecture) suit in a more general context of

the Langlands Conjecture for GL

An elliptic curve E deﬁned over the rational numbers can be given by an equation

= x

+ ax + b,wherea, b œ Z and the polynomial on the right hand side has 3

diﬀerent zeros in C.IfZ æ R is a homomorphism into a ring R, then the set of

R≠rational points of E is deﬁned as:

E(R)={(x, y) œ R

| y

= x

+ ax + b},

where the equation is considered as an equation with coeﬃcients in R (through the

given homomorphism from Z to R). If R is a ﬁeld, then E(R) has a structure of

an abelian group (see e.g. [W], 2.2).

The Hass e- We il L-function of the elliptic curve E is usually deﬁned in the following

way (see [W], 14.2), which has no clear relation to the Galois groups. One studies

the numbers of points on the curve E reduced modulo diﬀerent prime numbers,

that is, the numbers |E(Z/pZ)|,where|X| denotes the number of elements in

the set X. Now there is a diﬀerence between the primes. In general, the curve

E is still an elliptic curve when reduced modulo p, that is, considered over Z/pZ

through the natural homomorphism Z æ Z/pZ. Then we say that E has good

reduction modulo p. It is always the c ase when p does not divide the discriminant

≠16(4a

+ 27b

). The remaining primes behave in diﬀerent ways. We shall not

discuss their behavior but we call them “bad”. For any good prime, that is, such

that the reduction of E modulo p is an elliptic curve over Z/pZ,wedeﬁne

= p +1≠|E(Z/pZ)|

and we don’t discuss the deﬁnition of a

for bad primes (it is 0, 1 or ≠1 depending

on diﬀerent cases). Now the Hasse-Weil L-function of E is deﬁned as the following

inﬁnite product:

L(E,s)=

p bad

(1 ≠ a

≠s

)

≠1

p good

(1 ≠ a

≠s

+ p

1≠2s

)

≠1

It was proved by Hasse (1933) that L(E,s) is holomorphic for Ÿ(s) > 3/2 and

conjectured that the function has analytic continuation to the whole complex plane

and satisﬁes a functional equation relating its values at s and 2 ≠ s. A more exact

form of this relation was conjectured by Weil (1967) for a slightly modiﬁed function

L(E,s) (which is s ometime s denoted by L(E,s), while the original function deﬁned

by Hasse is called zeta function of E and denoted by ’(E,s), see [BG], Chap. 5).

It is far from being evident that the deﬁnition of the L-function of E can be given

in terms of Galois groups and their representations. In a moment, we describe

such a deﬁnition but ﬁrst let us note that it is not necessary for a formulation of

the Taniyama-Shimura-Weil Conjecture whose proof (in a special case ) resulted

in a proof of Fermat’s Last Theorem given by Wiles. However, the transition to

Galois representations deﬁned by elliptic curves is necessary as a key to the proof

Normat 3-4/2011 Juliusz BrzeziÒski 167

of the Conjecture. For us it is essential as an example showing that the Langlands

Conjecture is relevant in this case and that the validity of the Taniyama-Shimura-

Weil Conjecture supports the Langlands Conjecture. Elliptic curves play a role of

a bridge between the two sides of the equality (6) in the Langlands Conjecture.

First the left hand side. Let E be an e lliptic curve over Q with Hasse-Weil L-

function

(7) L(E,s)=

n=1

The Taniyama-Shimura-Weil Conjecture says that for any elliptic curve E over Q

the function

(8) Ï

(s)=

n=1

where q = e

2ﬁis

is a modular cusp form for 

(N) for some N (see Theore m 10

as an explanation of this notion). An elliptic curve with this property is called

modular. Thus the Taniyama-Shimura-Weil Conjecture says that every elliptic

curve over the rational numbers is m odular. As a notational convention, we denote

by L(Ï, s) the right hand side of (7) if Ï is given by the right hand s ide of (8). We

say that Ï

(s) is the q-expansion of the function L(E,s).

Andrew Wiles (with the cooperation of Richard Taylor) proved in 1994 that the

Taniyama-Shimura-Weil Conjecture is true for a special class of elliptic curves

over the rational numbers. This class contains the c urves y

= x(x ≠ a

)(x + b

where p>5 is a prime and a

+ b

= c

for some positive integers a, b, c.Such

elliptic curves are called Hellegouarch-Frey elliptic curves. It was proved by

Kenneth Rib e t in 1986 that if a Hellegouarch-Frey elliptic curve exists (that is,

the corresponding Fermat equation has a solution), it is not modular. Thus Wiles

proved that such curves can not exist, that is, there are no nontrivial solutions to

the Fermat equation a

+ b

= c

when p>5.

Let us note that the Taniyama-Shimura-Weil Conjecture was proved in its full

generality in 1999 by Christophe Breuil, Brian Conrad, Fred Diamond and Richard

Taylor.

As we pointed earlier, at the same time when Emil Artin introduced his L-functions,

Erich Hecke studied relations between holomorphic functions

Ï(s)=

n=1

,q= e

2ﬁis

,sœ C, ⁄s>0

(sometimes satisfying some extra conditions) and the corresponding functions

168 Juliusz BrzeziÒski Normat 3-4/2011

L(Ï, s)=

n=1

These investigations were continued by André Weil (1967) who generalized Hecke’s

results for SL

(Z) showing that if Ï(s) is a modular cusp form of weight k for a

group 

(N) and an eigenvector for all Hecke operators (on the vector space of

all modular cusp forms of weight k), then L(Ï, s) has all properties which very

often one expects (e.g. in the Hasse-Weil Conjecture or in the Artin Conjecture)–

it extends uniquely to a holomorphic function on the whole complex plane and

satisﬁes a functional equation relating its values at s and k ≠ s.

In his proof, Andrew Wiles used Galois representations G(Q/Q) æ GL

from elliptic curves. If E is an elliptic curve and ﬂ

: G(Q/Q) æ GL

representation described below, then the Artin L-function L(ﬂ

,s) is the same as

the Hasse-Weil L-function L(E,s).Insuchaway,wehave

L(ﬂ

,s)=L(E,s)=L(Ï

,s)

for the modular form Ï

corresponding to E. This is exactly the kind of result

which follows the prediction of the Langlands Conjecture for n =2. Thus the main

result of Wiles, which gave the ﬁnal proof of Fermat’s Last Theorem, was a proof of

(a special case of) the Shimura-Taniyama-Weil conjecture (which at the same time

gives validity of the Hasse-Weil Conjecture). We formulate this result in a form,

which takes into account the fact that the Shimura-Taniyama-Weil Conjecture is

proved in its full generality:

Theorem 10 For each elliptic curve E deﬁned over Q there exists a modular cusp

form Ï such that

L(E,s)=L(Ï, s).

More exactly, if L(E,s) is Hasse-Weil L-function of E and Ï

(s) the corresponding

q-expansion of it, then Ï = Ï

and there exists an integer N such that for all s œ C,

⁄s>0, we have

as + b

cs + d

=(cs + d)

(s) for every a, b, c, d œ Z,ad≠ bc =1and N | c,

and

≠1

= ±Ns

(s).

Normat 3-4/2011 Juliusz BrzeziÒski 169

The two conditions above deﬁne Ï

as a modular cusp form of weight 2 and level

N for the group 

(N) (this group of 2◊2 matrices is described in the last theorem

by the conditions on their elements a, b, c, d ).

What we need now is an explanation of the way from an elliptic curve E to a

Galois representation ﬂ

: G(Q/Q) æ GL

L(E,s).

Consider as before, an elliptic curve E deﬁned over Q. Using the group structure

deﬁned on the points (x, y) œ Q

of E, we consider the subgroups E[m] of E(Q)

consisting of elements whose orders divide m. It is well known that E[m]

≥

Z/mZ◊

Z/mZ (see [W], p. 79).

Now the automorphisms ‡ from the Galois group G(Q/Q) act on the group E[m]:

If (x, y) œ E(Q),then‡(x, y)=(‡(x),‡(y)). Fixing an isomorphism with Z/mZ ◊

Z/mZ and choosing a basis, we get a matrix representation of ‡ as an invertible

linear mapping. Thus, for every positive integer m, we get a representation:

ﬂ

: G(Q/Q) æ GL

(Z/mZ).

In particular, we ﬁx a prime number p and choose as m all its powers m = p

for

k =1, 2,.... A not complicated construction shows that we can choose bases for

Z/p

Z ◊ Z/p

Z in such a way that matrices M

corresponding to ﬂ

are equal

modulo p

for all k =1, 2,...:

= M

k+1

(mod p

Such a sequence of matrices deﬁnes a matrix over the p-adic integers Z

(see the

Appendix on p-adic numbers ) and in this way a representation:

ﬂ

: G(Q/Q) æ GL

The ring Z

(of characteristic 0) has a ﬁeld of quotients Q

. Its algebraic closure

can be embedded into the complex numbers. In fact, the (ultra) metric on

the ﬁeld of p-adic numbers can b e uniquely extended to Q

and further to an

algebraically closed ﬁeld C

which is complete with respect to the extended metric

and isomorphic, as a ﬁeld, to the complex numbers. If we ﬁx such an isomorphism,

we get a representation:

ﬂ

: G(Q/Q) æ GL

(C).

Notice that we use the same notation for this representation, since we only want to

point out a possibility to replace C

by C in order to see the relevance of this case

in the general context of Langlands Conjecture for GL (in fact, it is not necessary

to return to complex numbers in order to formulate what follows below). Thus, we

have obtained a representation ﬂ

= ﬂ

deﬁned by the elliptic curve E. What we

170 Juliusz BrzeziÒski Normat 3-4/2011

need now is to show that the Artin L-function corresponding to the represe ntation

ﬂ

is exactly the same as the Hasse-Weil L-function L(E,s).

Denote F

= Z/pZ. Let p be a prime such that E has good reduction modulo it.

Then we have the geometric Frobenius endomorphism Fr

of E(F

) deﬁned by

(x, y)=(x

). In the Galois group G(Q/Q), there is a Frobenius element

whose image in G(F

) is the Frobenius automorphism x ‘æ x

. Essentially

this is the Frobenius automorphism discussed earlier, but since now, we consider

the Galois group G(Q/Q), we have to modify somewhat the deﬁnition (taking

restrictions to the ﬁnite extension K of the rational numbers). Now, we have (see

[BG], Chap. 5):

Theorem 11 Let q be a prime such that E has a good reduction modulo q and

q ”= p. Then we have det(ﬂ

(Fr

)) = q and

tr(ﬂ

(Fr

)) = 1 + q ≠|E(F

Thus the L-function corresponding to the representation ﬂ

deﬁned according to

the Artin’s deﬁnition will be

L(E,s)=

det

≠ ﬂ

(Fr

≠s

≠1

which taking into account Theorem 11 gives

det(

≠ ﬂ(Fr

≠s

)=1≠tr(ﬂ

(Fr

≠s

+det(ﬂ

(Fr

))q

≠2s

=1≠a

≠s

≠2s+1

Hence,

L(ﬂ

,s)=

(1 ≠ a

≠s

+ q

1≠2s

)

and disregarding “the bad primes”, we get exactly the same Hasse–Weil L-function

of the elliptic curve E.

12 A few comments on the Langlands program

As we noted earlier, the Langlands Conjecture for GL

is only one of several state-

ments, which form the whole Langlands Program. Moreover, the group GL

can

We use the te rm “endomorphism” and not “ automorphism” since Fr

is not invertible as a

mapping of the elliptic curve. We say “geometric” since it is deﬁned on points of a curve and not

as before on a ﬁeld extension.

Normat 3-4/2011 Juliusz BrzeziÒski 171

be replaced by any group in a big class of algebraic groups for which the Program

is formulated. Even if there existed only a few results conﬁrming the Langlands

Program, when it was formulated, the new developments show more and more in

support of it. It is not restricted to the algebraic number ﬁelds. There are natural

versions of the conjectures in many other cases, which usually are studied in con-

nection with other mathematical problems when there is a possibility to formulate

analogical questions. Thus, there are versions of the Program for local and global

ﬁelds, ﬁnite ﬁelds, function ﬁelds of algebraic varieties over complex numbers and

others. The part of the Program related to function ﬁelds of algebraic varieties is

usually called Geometric Langlands. Many mathematicians working in these very

central, active and diﬃcult ﬁelds of mathematics contributed with results support-

ing the Program, which probably will take centuries to fulﬁll and develop. Just an

analogue of the class ﬁeld theory for all Galois extensions (not only abelian) is a

tremendous task!

As we noted before, a fundamental question was how to replace Dirichlet charac-

ters over the rational numbers (or Hecke characters over arbitrary algebraic num-

ber ﬁelds) on the right hand side of the equation (6). In order to formulate the

conjectures in full generality, Langlands deﬁned a new kind of groups (L-groups),

L-functions and special group representations. This is a very extensive theoretical

material, which of course we are not able to discuss here. There are several good

review articles and collections of such related to this subject, notably, [Kn],[G],[BG].

What we shall try to explain is how to ﬁnd a context in which diﬀerent results such

as class ﬁeld theory or Shimura-Taniyama-Weil Conjecture (Hasse-Weil Conjecture)

can be studied. The groups involved are GL

. These groups can be considered over

arbitrary commutative rings. In the special case, which is the closest to the classic al

class ﬁeld theory, they are linear groups GL

over so called adele ri ngs.Inthe

simplest case, which we discuss, these groups are the linear groups GL

) over

the adele ring A

of the rational numbers. The adele ring A

of Q involves a kind

of product of all possible completions of Q with respect to both the archimedean

metric (the absolute value) and all non-archimedean (ultra-)metrics corres ponding

to all prime numbers (see the Appendix). Thus the real numbers are involved as

the “inﬁnite component” of GL

) – the completion with respect to the usual

absolute value on Q. This completion is the group GL

(R). Langlands studies the

representations of the groups GL

) on s ome particular function spaces on the

quotients GL

)/GL

(Q)Z

). Here the group GL

(Q) of n◊n matrices over

the rational numbers is embedded diagonally in the product deﬁning the group

), and Z

) is the c enter of this group. In the case n =1,wehave

representations of the group GL

)=A

on the complex numbers, which in fact

are characters of the group GL

)/GL

(Q)=A

, that is, homomorphisms

æ C

. It appears that here “automorphic cuspidal representations” are

closely related to the usual Dirichlet characters. A detailed explanation of this

relation can be found in [B], pp. 258-259.

In the case n =2, we get representations of GL

) on a particular function space

on GL

)/GL

(Q)Z

). It is shown in [GGP-S], Appendix to §4 of Chapter 3,

how to lift the classical modular cusp forms, which correspond to representations of

the inﬁnite component GL

(R) of GL

) (modulo congruence subgroups 

(N)

172 Juliusz BrzeziÒski Normat 3-4/2011

as in section 11) to representations of the groups GL

). Thus, the “automorphic

cuspidal representations” appear in this case as modular cusp forms considered in

the last section.

After 1970 several important mathematical achievements were related to the Lang-

lands Program. One of the ﬁrst positive results for n>1 was concerned with the

case n =2, that is, with representations ﬂ : G(Q/Q) æ GL

(C). Pierre Deligne

and Jean-Pierre Serre showed that a special class of modular cusp forms of weight

1 for the group 

(N) correspond to a special class of irreducible representations

of dimension 2 in accordance with the Langlands Conjecture for GL

. Recall that

Wiles result is also related to two-dimensional Galois representations over the ratio-

nal numbers. Several other important results are proved in this case, but the general

case related to the representations ﬂ : G(Q/Q) æ GL

solved.

Robert Langlands proved several of his conjectures related to the ﬁelds of real and

complex numbers. George Lusztig proved corresponding results over ﬁnite ﬁelds.

Laurent Laﬀorgue proved Langlands conjectures for the general linear group GL

(K)

over global function ﬁelds K (1998), which gave a generalization of earlier results of

Vladimir Drinfeld for n =2(1974). Philip Kutzko (1980) proved Langlands conjec-

tures for GL

(K) over local ﬁelds. Gérard Laumon, Michael Rap oport, and Ulrich

Stuhler (1993) proved the Langlands conjectures for the groups GL(n, K) when

K is a ﬁeld of a nonzero characteristic. Richard Taylor and Michael Harris (2001)

proved the Langlands conjectures for GL

(K) for local ﬁelds K of characteristic

0. This was proved also by Guy Henniart (2000). A short time ago, Ngo Bao Chau

proved a closely related result which plays an important role in Langlands Program

(the so-called “Fundamental Lemma”).

Several of these achievements were rewarded with diﬀerent international prizes

in mathematics including several Fields Medals (Vladimir Drinfeld, Laurent Laf-

forgue, Ngo Bao Chau) and probably, many more will earn this greate st mathe-

matical honour through new achievements related to the Langlands Program for

many years in the future.

APPENDIX

A.1 Fermat’s t heorem on two squares.

Here we give a proof of the Fermat theorem 1 saying that a prime p is a sum of

two integer squares if and only if p =2or p = 1 (mod 4).

The “only if” part is very easy to prove. In fact, if p is and odd prime and a sum

of two squares, then one of them must by even and the other one odd. A square

of an even number gives residue 0 modulo 4, and a square of an odd number gives

residue 1 (if x =2k +1,thenx

= 4(k

+ k)+1). Thus p as a sum of an even

and an odd square must give residue 1 modulo 4. Hence, primes giving residue 3

modulo 4 can not be sums of two integer squares.

Normat 3-4/2011 Juliusz BrzeziÒski 173

Conversely, we have to prove that if p = 1 (mod 4),thenp is a sum of two integer

squares. We already know from Lemma 1 that the equation x

+1 = 0 has a

solution modulo p, so there is an integer x such that p | x

+1=(x + i)(x ≠ i) in

Z[i]. This means that (x + i)(x ≠ i) œÈpÍ.IfÈpÍ is a prime ideal, then x + i œÈpÍ

or x ≠ i œÈpÍ. But it means that x + i = p(a + bi) or x ≠ i = p(a ≠ bi) (conjugate

the ﬁrst equality). This is of course impossible since p - 1. Thus the ideal ÈpÍ is not

prime, and p is reducible, that is, p =(x + yi)(z + ti) in Z[i],whereneitherx + yi

nor z + ti is a unit. Taking the square of the absolute value, we get (as before in

section 4) that p = x

+ y

is a sum of two integer squares.

A.2 Legendre symbol

Here we gather a few properties of the Legendre symbol, which we need in the text.

Let p be an odd prime number and a an integer. The Legendre symbol

gives

information about the existence of solutions to quadratic equations modulo primes.

If p - a, then the symbol is equal 1 when the equation x

= a has a solution modulo

p, and ≠1 when such a solution does not exist. We deﬁne the value of the symbol

equal 0 if p | a. Since the mapping x ‘æ x

is a homomorphism of the group (Z/pZ)

with p ≠ 1 elements and the kernel of this homom orphism consists of exactly two

residues ±1 (the equation x

=1has two solutions ±1 in Z/pZ), there are exactly

p≠1

nonzero residues modulo p, which are squares. This means that the equation

= a (mod p) is solvable for e xactly

p≠1

residues a. This observation gives what

is known as the Euler criterion:

= a

p≠1

(mod p).

In fact, the nonzero residues in Z/pZ form a group of order p ≠ 1 so a

p≠1

=1for

each nonzero a (this is Fermat’s Little Theorem). Hence the equation

(9) x

p≠1

≠ 1=(x

p≠1

≠ 1)(x

p≠1

+ 1) = 0

has exactly p≠1 solutions. If x

= a has a solution, then a

p≠1

=(x

)

p≠1

= x

p≠1

1. Thus those a for which the Legendre symbol equals 1 are exactly the zeros of

the ﬁrst factor in (9). Hence the remaining

p≠1

residues are the zeros of the second

factor of (9). For them the Legendre symbol is equal ≠1 and a

p≠1

= ≠1.

Notice that Euler’s criterion shows that

as well as gives the equality

≠1

=(≠1)

p≠1

, which we used in section 5.

174 Juliusz BrzeziÒski Normat 3-4/2011

A.3 Quadratic reciprocity law.

As an application of the Kronecker-Weber Theorem, we prove the law of quadratic

reciprocity:

Theorem 12 Let p and q be two diﬀerent odd primes. Then

=(≠1)

p≠1

q≠1

Proof. Let p and q be two diﬀerent odd prime numbers. Consider the cyclotomic

ﬁeld K = Q(Á

),whereÁ

=1and Á

”=1. Since the Galois group G = G(K/Q)=

(Z/pZ)

is cyclic of (even) order p ≠ 1,theﬁeldK contains exactly one quadratic

subﬁeld M, which corresp onds the subgroup of index 2 in G. This subgroup is G

the subgroup of all squares of the elements of G, since the homomorphism g ‘æ g

of G has the kernel consisting of the two residues satisfying the equation g

=1,

that is, ±1. Its image is H = G

so it is a subgroup of index 2.

The only quadratic subﬁeld of K is M = Q(

),wherep

=(≠1)

p≠1

p.This

fact, known since Gauss published “Disquisitiones Arithmetica”, can be proved in

several diﬀerent ways. One method is to refer to Theorem 5 noting that the only

prime number ramiﬁed in Q(

) is just p and the same prime number is ramiﬁed

in K.

Now the prime number q is s plit in M if and only if the polynomial x

≠ p

is a

product of two linear factors modulo q, that is, if and only if

=1.

At the same time, the Frobenius automorphism Fr

=1on M if and only if Fr

is a square in G(K/Q), that is, the residue q is a square in (Z/pZ)

, which means

=1.

Hence

which is only just another (typographic) form of the quadratic reciprocity:

(≠1)

p≠1

≠1

p≠1

=(≠1)

p≠1

q≠1

Normat 3-4/2011 Juliusz BrzeziÒski 175

A.4 p-adic numbers.

It is well known that every real number is a limit of a sequence of rational num-

bers with respect to the distance deﬁned by the absolute value. For example, if

– = a

...,wherea

is the integer part of – and a

for k =1, 2,... are its

consecutive decimal digits, then – =lim

kæŒ

–

,where

–

= a

100

+ ···+

We say that the real numbers R are obtained as a completion of the rational

numbers with respect to the metric deﬁned by the absolute value, that is, d

(a, b)=

|a ≠ b| for a, b œ Q. But there are other metrics on the rational numbers – every

prime number p deﬁnes a metric if we deﬁne d

(a, b)=p

≠k

if a ≠b = p

,where

r, s are integers and p - rs. It is easy to check that Q with respect to d

is a metric

space. The essential diﬀerence between d

and d

is that the last metrics satisfy a

very strong form of the triangle inequality:

(a, b) Æ d

(a, c)+d

(b, c) while d

(a, b) Æ max(d

(a, c),d

(b, c)).

for any a, b, c œ Q.Themetricd

is called archimedean, while d

for prime numbers

p are called discrete metrics (or non-archimedean). But one can also complete the

rational numbers with respect to these metrics. The resulting limits of sequence s

of rational numbers are called p-adic numbers and are denoted by Q

. A typical p-

adic number can be obtained in the following way. Every integer a can be uniquely

written in the basis p as a sum

a = a

+ a

p + ···+ a

where 0 Æ a

<p(that is, the digits in the basis p are 0, 1,...,p≠ 1). A p-adic

integer – can be uniquely presented as a convergent series with respect to the

metric d

, that is,

– = a

+ a

p + ···+ a

+ ··· ,

where 0 Æ a

<p, and if

–

= a

+ a

p + ···+ a

then d

(–

,–

) æ 0 when k, l æŒ(it is an easy exercise to show that the last

condition c an be replaced by d

(–

,–

k+1

) æ 0 when k æŒ). The p-adic integers

are denoted by Z

and it is possible to show that with respect to natural extension

of addition and multiplication from Z they form a ring. Now the ﬁeld of p-adic

numbers Q

can be deﬁned as the quotient ﬁeld of Z

.If– is a p-adic integer, then

– = p

Á, r Ø 0,where

176 Juliusz BrzeziÒski Normat 3-4/2011

Á = a

+ a

p + ···+ a

+ ···

has a

”=0. It is easy to check that such a p-adic integer has an inverse in the ring

. Hence each p-adic number in Q

can be uniquely written as – = p

Á,wherek

is an integer.

It appears that the metrics d

and d

for prime numbers p are essentially all

possible on Q (when suitable notion of equivalent metrics is deﬁned). A global

object used in connection with group representations in number theory is the adele

ring of Q which makes it possible to study simultaneously all metrics. In fact, their

roles are often equally es se ntial and it is very convenient to have a possibility to

treat them equally. The adele ring Q

is deﬁned as the product of all Q

(it is

convenient to denote R = Q

), that is, the set of all sequences (–

),where–

œ Q

and –

œ Z

for all p with possibly a ﬁnite number of exceptions. It is easy to check

that Q

is a ring – the adele ring of Q.

A.5 Why “classes” in class ﬁeld theory?

Sometimes, each ideal can be generated by one number, that is, can be given in

the form I = È–Í.InZ or Z[i] all ideals can be represented in this way. We say

that these rings are principal ideal rings. In fact, the ring of integers in any

algebraic number ﬁeld K is a principal ideal ring if and only if the factorization

of its elements into irreducibles is unique. But this property is rather exceptional.

A measure of its failure is the size of the class group of the ring O

. We say that

two ideals I, J of this ring belong to the same class if there are numbers –, — œO

such that –I = —J . Denoting the class of I by [I], we can deﬁne multiplication of

classes in the natural way: [I][J]=[IJ]. It is possible to prove that classes of the

ideals of O

form a ﬁnite group C(O

) whose unit is the class of O

. The order

of this group is called the class number of O

and will be denoted by h(O

).The

class number h(O

)=1if and o nly if all ideals in O

are principal, which means

that O

has unique factorization. A usual way to deﬁne the class group C(O

)

is the quotient of the group I

of all ideals of K (including so called fractional

ideals) by the subgroup P

of principal (fractional) ideals.

As we noted in section 4, the unique factorization fails in the ring Z[

≠5]. In fact,

the ideal p = È3, 2+

≠5Í is not principal. Assuming that p can be generated by

only one number – = a + b

≠5, a, b œ Z, we get È3, 2+

≠5Í = Èa + b

≠5Í.This

gives 3=(a + b

≠5)(c + d

≠5) and 2+

≠5=(a + b

≠5)(c

+ d

≠5). Taking

the absolute values, we ge t a

+5b

=1or 3 or 9 and each case easily results in a

contradiction. In fact, it is possible to prove that the class number of Z[

≠5] is 2,

that is, every ideal is principal or is of form –p,where– œ Q[

≠5].

One of the main problems in number theory is to study class groups of the rings

of integers in algebraic number ﬁelds. The knowledge of class groups is decisive

in many interesting and important problems. One of the fundamental results in

class ﬁeld theory is a beautiful relation between Galois groups and class groups of

algebraic number ﬁelds. This relation explains in fact the source of the name of the

theory – class ﬁeld theory. The close relation between class groups of the algebraic

Normat 3-4/2011 Juliusz BrzeziÒski 177

integers in algebraic number ﬁelds and Galois groups of the abelian ﬁeld extensions

of these ﬁelds is manifested in many theorems. Let us mention one, probably most

distinct, which relates so called Hilbert class ﬁeld to the class group. Let K be

an algebraic number ﬁeld. The Hilbert class ﬁeld of K is a Galois extension H of

K such that the Galois group G(H/K) is isomorphic to the class group of O

means of a mapping deﬁned by the Artin symbol (see the end of section 8):

H/K

: I

æ G(H/K),

which gives a surjective map onto the Galois group and whose kernel is the sub-

group of the group I

generated by the principal ideals in H, that is, the induced

isomorphism of the quotient is just the isomorphism of the class group C(O

) with

G(H/K). Let us mention that the Hilbert class ﬁeld of K is the maximal abelian

unramiﬁed extension of this ﬁeld.

Acknowledgment. The author would like to thank David Cox for many helpful

comments and suggestions.

References

[BG] Bernstein, J., Gelbart, S. (Editors): Introduction to the Langlands program,

Birkhäuser, Boston, 2003.

[B] Bump, D.: Automorphic Forms and Representations, Cambridge University

Press, (1997).

[C] Cox, M.: Primes of the form x

+ ny

, Fermat, Class Field Theory, and

Complex Multiplication, John-Wiley &Sons, Inc., New York 1989.

[G] Gelbart, M.: An Elementary Introduction to the Langlands Program, Bull.

Am. Math. Soc. 10,177–219(1984).

[GGP-S] Gelfand, I.M., Graev, M.I., Piatetski-Shapiro, I.: Representation theory and

automorphic functions, W. B. Saunders Company, Philadelphia, (1969).

[IK] Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Math. Soc.,

Colloquium Publications vol. 53,Providence R.I. 2004.

[Kn] Knapp, A.W.: Introduction to the Langlands Program, Proc. Symp. Pure

Math. 61,245–302(1997).

[Ko] Koch, H.: Number Theory, Algebraic Numbers and Functions, Graduate

Studies in Mathematics, vol. 24, American Math. Soc., Providence R.I. 2000.

[N] Nagell, T.: Introduction to Number Theory, AMS Chelsea Publishing,

Providence R.I., (1964).

[W] Washington, L.C.: Elliptic Curves, CRC Press, Boca Raton FL. 2008.