Algebraic number theory, serge lang 1

Graduate Texts in Mathematics

11 0

Editorial Board
F. W. Gehring P. R. Halmas (Managing Editor)
C. C. Moore


Graduate Texts in Mathematics
A Selection
60

ARNOLD. Mathematical Methods in C1assical Mechanics.

61
62

WHITEHEAD. Elements of Homotopy Theory.
KARGAPOLOV/MERZIJAKOV. Fundamentals of the Theory of Groups.

63

BOLLABAs. Graph Theory.

64

EDWARDS. Fourier Series. Vol. I. 2nd ed.

65

66

WELLS. Differential Analysis on Complex Manifolds. 2nd ed.
W ATERHOUSE. Introduction to Affine Group Schemes.

67

SERRE. Local Fields.

68
69

WEIDMANN. Linear Operators in Hilbert Spaces.
LANG. Cyclotomic Fields 11.

70
71

MASSEY. Singular Homology Theory.
FARKAS/KRA. Riemann Surfaces.

72
73

STILLWELL. C1assical Topology and Combinatorial Group Theory.
HUNGERFORD. Algebra.

74
75

DAVENPORT. Multiplicative Number Theory. 2nd ed.
HOCHSCHILD. Basic Theory of Aigebraic Groups and Lie Aigebras.

76

IITAKA. Aigebraic Geometry.

77
79

HEcKE. Lectures on the Theory of Aigebraic Numbers.
WALTERS. An Introduction to Ergodic Theory.

80

ROBINSON. A Course in the Theory of Groups.

81
82

FORSTER. Lectures on Riemann Surfaces.
BOTT/Tu. Differential Forms in Aigebraic Topology.

83
84

WASHINGTON. Introduction to Cyclotomic Fields.
IRELAND/RoSEN. A C1assicallntroduction to Modern Number Theory.

85

EDWARDS. Fourier Series: Vol. 11. 2nd ed.

86
87

VAN LINT. Introduction to Coding Theory.
BROWN. Cohomology of Groups.

88
89

PIERCE. Associative Aigebras.
LANG. Introduction to Aigebraic and Abelian Functions. 2nd ed.

91
92

BEARDON. On the Geometry of Discrete Groups.
DIESTEL. Sequences and Series in Banach Spaces.

93
94

DUBROVIN/FoMENKO/NoVIKOV. Modern Geometry-Methods and Applications Vol. I.
WARNER. Foundations of Differentiable Manifolds and Lie Groups.

95
96

SHIRYAYEV. Probability, Statistics, and Random Processes.
CONWAY. A Course in Functional Analysis.

97

KOBLITZ. Introduction to Elliptic Curves and Modular Forms.

98
99

BRÖCKER/tom DIECK. Representations of Compact Lie Groups.
GROVE/BENSON. Finite Reflection Groups. 2nd ed.

100
101

BERG/CHRISTENSEN/RESSEL. Harmonic Analysis on Semigroups: Theory of positive definite
and related functions.
EDWARDS. Galois Theory.

102
106

VARADARAJAN. Lie Groups, Lie Aigebras and Their Representations.
SILVERMAN. The Arithmetic of Elliptic Curves.

107

OLVER. Applications of Lie Groups to Differential Equations.

Serge Lang

Alge braic N um ber
Theory

Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo


Serge Lang
Department of Mathematics
Yale U niversity
New Haven, CT 06520

U.S.A.
Editorial Board
P. R. Halmos

F. W. Gehring

C.C. Moore

M anaging Editor
Department of
M.athematics
U niversity of Santa Clara
Santa Clara, CA 95053
U.S.A.

Department of
Mathematics
University of l\Iichigan
Ann Arbor, MI 48109
U.S.A.

Department of
Mathematics
University of California
at Berkeley
Berkeley, CA 94720
U.S.A.

AM.S Classitications: 1065. 1250
With 7 Illustrations
Library of Congress Cataloging in Publication Data
Lang, Serge
Aigebraic number theory.
(Graduate texts in mathematics; 110)
Bibliography: p.
Includes index.
1. Aigebraic number theory. 2. Class field theory.
I. Title. H. Series.
QA247.L32 1986
512'.74
86-6627
Originally published in 1970
Reading, Massachusetts.

© by Addison-Wesley Publishing Company, Inc.,

© 1986 by Springer-Verlag New York Inc.
Softcover reprint ofthe hardcover 1st edition 1986
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag. 175 Fifth Avenue. New Y ork,
New York 10010. U.S.A.

9 8 7 654 3 2 1

ISBN 978-1-4684-0298-8
ISBN 978-1-4684-0296-4 (eBook)
DOI 10.1007/978-1-4684-0296-4


Foreword
The present book gives an exposition of the classical basic algebraic
and analytic number theory and supersedes my Algebraic Numbers,
including much more material, e.g. the class field theory on which I make
further comments at the appropriate place later.
For different points of view, the reader is encouraged to read the collection of papers from the Brighton Symposium (edited by Cassels-Frohlich),
the Artin-Tate notes on class field theory, Weil's book on Basic Number
Theory, Borevich-Shafarevich's Number Theory, and also older books like
those of Weber, Hasse, Hecke, and Hilbert's Zahlbericht. It seems that
over the years, everything that has been done has proved useful, theoretically or as examples, for the further development of the theory. Old,
and seemingly isolated special cases have continuously acquired renewed
significance, often after half a century or more.
The point of view taken here is principally global, and we deal with
local fields only incidentally. For a more complete treatment of these,
cf. Serre's book Corps Locaux. There is much to be said for a direct global

approach to number fields. Stylistically, I have intermingled the ideal
and idelic approaches without prejudice for either. I also include
two proofs of the functional equation for the zeta function, to acquaint
the reader with different techniques (in some sense equivalent, but in
another sense, suggestive of very different moods). Even though areader
will prefer some techniques over alternative ones, it is important at least
that he should be aware of all the possibilities.
New York
June 1970

SERGE LANG


Prerequisites
Chapters I through VII are self-contained, assuming only elementary
algebra, say at the level of Galois theory.
Some of the chapters on analytic number theory assume so me analysis.
Chapter XIV assumes Fourier analysis on locally compact groups. Chapters XV through XVII assume only standard analytical facts (we even
prove so me of them), except for one aUusion to the Plancherel formula in
Chapter XVII.
In the course of the Brauer-Siegel theorem, we use the conductordiscriminant formula, for which we refer to Artin-Tate where a detailed
proof is given. At that point, the use of this theorem is highly technical,
and is due to the fact that one does not know that the zeros of the zeta
function don't occur in a smaU interval to the left of 1. If one knew this,
the proof would become only a page long, and the L-series would not be
needed at aU. We give Siegel's original proof for that in Chapter XIII.
My Algebra gives more than enough background for the present book.
In fact, Algebra already contains a good part of the theory of integral
extensions, and valuation theory, redone here in Chapters land II.
Furthermore, Algebra also contains whatever will be needed of group

representation theory, used in a couple of isolated instances for applications of the class field theory, or to the Brauer-Siegel theorem.
The word ring will always mean commutative ring without zero divisors
and with unit element (unless otherwise specified).
If K is a field, then K* denotes its multiplicative group, and K its
algebraic closure. Occasionally, a bar is also used to denote reduction
modulo a prime ideal.
We use the 0 and 0 notation. If I, gare two functions of a real variable,
and g is always ~ 0, we write 1 = O(g) if there exists a constant C > 0
such that I/(x) I ~ Cg(x) for all sufficiently large x. We write 1 = o(g) if
limz--+oo/(x)/g(x) = O. We write/"" g if lim", ___",/(x)/g(x) = 1.

vii


Contents
Part One
General Basic Theory
CHAPTER

I

Algebraic In tegers

1.
2.
3.
4.
5.
6.
7.

8.

Localization
Integral closure
Prime ideals .
Chinese remainder theorem
Galois extensions
Dedekind rings
Discrete valuation rings
Explicit factorization of a prime

3
4
8
11
12
18
22
27

CHAPTER

II

COInpletions

1.
2.
3.
4.

5.

Definitions and completions
Polynomials in complete fields
Some filtrations .
Unramified extensions
Tamely ramified extensions

31
41

45
48
51

CHAPTER

III

The Different and Discriminant

1.
2.
3.

57

Complementary modules
The different and ramification
The discriminant

62
64

IX


x

CONTENTS
CHAPTER

IV

Cyclotomic Fields
l.

Roots of unity

71

2. Quadratic fields
3. Gauss sums
4. Relations in ideal classes

76
82
96

CHAPTER

V

ParalIelotopes
l.

The product formula

99
110
116
119

2. Lattice points in parallelotopes
3. A volume computation
4. Minkowski's constant

CHAPTER

VI

The Ideal Function
l.

Generalized ideal classes

2. Lattice points in homogeneously expanding domains
3. The number of ideals in a given class .

CHAPTER

123
128
129

VII

Ideles and Adeles
l.

2.
3.
4.
5.
6.

Restricted direct products
Adeles.
Ideles
Generalized ideal class groups; relations with idele classes
Embedding of
in the idele classes
Galois operation on ideles and idele classes

k:

CHAPTER

137
139

140
145
151
152

VIII

Elementary Properties of the Zeta Function and L-series
l.

Lemmas on Dirichlet series .

2. Zeta function of a number field
3. The L-series
4. Density of primes in arithmetic progressions.

155
159
162
166


CONTENTS

xi

PartTwo
Class Field Theory
CHAPTER

IX

Norm Index Computations
1.
2.
3.
4.
5.
6.

Algebraie preliminaries .
Exponential and logarithm functions
The loeal norm index
A theorem on units .
The global eyelie norm index
Applieations .

CHAPTER

179
185
187
189
192
194

X

The Artin Symbol, Reciproeity Law, and Class Field Theory
1.

2.
3.

Formalism of the Artin symbol
Existenee of a eonduetor for the Artin symbol
Class fields

CHAPTER

197
200
206

XI

The Existenee Theorem and Loeal Class Field Theory
1.
2.
3.
4.
5.
6.

Reduetion to Kummer extensions
Proof of the existenee theorem .
The eomplete splitting theorem
Loeal class field theory and the ramifieation theorem
The Hilbert class field and the prineipal ideal theorem
Infinite divisibility of the universal norms

CHAPTER

213
215
217
219

224

225

XII

L-series Again

1.
2.
3.

The proper abelian L-series
Artin (non-abelian) L-series
Indueed eharaeters and L-series eontributions

229
232
236


xii

CONTENTS

Part Three
Analytic Theory
CHAPTER XIII
Functional Equation of the Zeta Function, Hecke's Proof
1.

2.
3.
4.
5.

The Poisson summation formula
A special computation
Functional equation .
Application t6 the Brauer-Siegel theorem
Applications to the ideal function .

245
250
253
260
262

CHAPTER XIV
Functional Equation, Tate's Thesis
1.

2.

3.
4.
5.
6.
7.
8.

Local additive duality
Local multiplicative theory .
Local functional equation
Local computations
Restricted direct products
Global additive duality and Riemann-Roch theorem
Global functional equation
Global computations

276
278
280
282
287
289
292
297

CHAPTER XV
Density of Primes and Tauberian Theorem
1.

The Dirichlet integral

2. Ikehara's Tauberian theorem
3. Tauberian theorem for Dirichlet se ries
4. N on-vanishing of the L-series
5. Densities

303
304
310
312
315

CHAPTER XVI
The Brauer-Siegel Theorem
1.

An upper estimate for the residue .

2. A lower bound for the residue
3. Comparison of residues in normal extensions
4. End of the proofs
Appendix: Brauer's lemma

322
323
325
327
328

CONTEN'rs
CHAPTER

xiii

XVII

Explicit Formulas
1.

2.
3.
4.
5.

Weierstrass factorization of the L-series
An estimate for A'/ A.
The basic sum
Evaluation of the sum: First part
Evaluation of the sum: Second part

331
333
336
338
340

Bibliography

351

Index.

353


PART ONE
BASIC THEORY


CHAPTER I

Algebraic Integers
This chapter describes the basic aspects of the ring of algebraic integers
in a number field (always assumed to be of finite degree over the rational
numbers Q). This includes the general prime ideal structure.
Some proofs are given in a more general context, but only when they
could not be made shorter by specializing the hypothesis to the concrete
situation we have in mind. It is not our intention to write a treatise on
commutative algebra.

§1. Localization
Let A be a ring. By a multiplicative subset of A we mean a subset
containing 1 and such that, whenever two elements x, y lie in the subset,
then so does the product xy. We shall also assume throughout that 0 does
not lie in the subset.
Let K be the quotient field of A, and let S be a multiplicative subset
of A. By S-1 A we shall denote the set of quotients xis with x in A and
s in S. It is a ring, and A has a canonical inclusion in S-1 A.
If M is an A-module contained in some field L (containing K), then

S-1M denotes the set of elements vls with v E M and SES. Then S-1M
is an S-1 A-module in the obvious way. We shall sometimes consider
the case when M is a ring containing Aassubring.
Let p be a prime ideal of A (by definition, p ~ A). Then the complement of p in A, denoted by A - p, is a multiplicative subset S = S~ of A,
and we shall denote S-1 A by A~.
A local ring is a ring which has a unique maximal ideal. If 0 is such a
ring, and m its maximal ideal, then any element x of 0 not lying in m
must be a unit, because otherwise, the principal ideal xo would be contained in a maximal ideal unequal to m. Thus m is the set of non-units
of o.
3


4

[1, §2]

ALGEBRAIC INTEGERS

The ring A p defined above is a loeal ring. As ean be verified at onee,
its maximal ideal mp eonsists of the quotients xis, with x in lJ and s in A
but not in lJ.
We observe that mp n A = lJ. The inclusion :J is clear. Conversely,
if an element y = xis lies in mp n A with xE lJ and SES, then x = sy E lJ
and s El lJ. Hence y E lJ.
Let A be a ring and S a multiplicative subset. Let a' be an ideal of
S-IA. Then
a' = S-I(a' n A).
The inclusion :J is clear. Conversely, let xE a'. Write x = als with
some a E A and SES. Then sx E a' n A, whence x E S-I(a' n A).
Under multiplieation by S-I, the multiplicative system of ideals of A

is mapped homomorphieally onto the multiplieative system of ideals of
S-1 A. This is another way of stating what we have just proved. If a
is an ideal of A and S-l a is the unit ideal, then it is elear that anS is
not empty, or as we shall also say, a meets S.

§2. Integral closure
Let A be a ring and x an element of some field L containing A. We
shall say that x is integral over A if either one of the following conditions
is satisfied.
INT 1. There exists a finitely genera ted non-zero A-module Me L such
that xM C M.
INT 2. The element x satisfies an equation

with coefficients ai E A, and an integer n
will be called an integral equation.)

~

1. (Such an equation

The two eonditions are actually equivalent. Indeed, assume INT 2.
The module M generated by 1, x, ... , x n - 1 is mapped into itself by the
element x. Conversely, assume there exists M = (VI, ••• , vn ) such that
xM C M, and M 7fI= O. Then

with coefficients aij in A. Transposing

XVI, .•• , XV n

to the right-hand side

[1, §2]

5

INTEGRAL CLOSURE

of these equations, we conclude that the determinant
x - an

is equal to O. In this way we get an integral equation for x over A.
Proposition 1. Let A be a ring, K its quotient field, and x algebraic over
K. Then there exists an element c ;t!= 0 of A such that cx is integral over A.
Proof. There exists an equation

with ai E A and an ;t!= O. Multiply it by a~-I. Then
(anx)n

+ ... + aoa~-I =

0

is an integral equation for anx over A.
Let B be a ring containing A. We shall say that B is integral over A
if every element of B is integral over A.
Proposition 2. If B is integral over A and finitely generated as an
A-algebra, then B is a finitely generated A-module.
Proof. We may prove this by induction on the number of ring generators, and thus we may assume that B = A[x] for so me element x integral over A. But we have al ready seen that our assertion is true in that
case.

Proposition 3. Let A C B C C be three rings. If B is integral over A
and C is integral over B, then C is integral over A.
Proof. Let x E C. Then x satisfies an integral equation

with bi E B. Let BI = A[b o, ... ,bn-d. Then BI is a finitely generated
A-module by Proposition 2, and BI[x] is a finitely generated BI-module,
whence a finitely generated A-module. Since multiplication by x maps
BI[x] into itself, it follows that x is integral over A.
Proposition 4. Let A eBbe ttW rings, and B integral over A. Let u
be a homomorphism of B. Then u(B) is integral over u(A).


6

ALGEBRAIC INTEGERS

[1, §2]

Proof. Apply u to an integral equation satisfied by any element x of B.
It will be an integral equation for u(x) over u(A).

The above proposition is used frequently when u is an isomorphism
and is particularly useful in Galois theory.
Proposition 5. Let A be a ring contained in a field L. Let B be the set
of elements of L tchich are integral over A. Then B is a ring, called the
integral closure of A in L.
Proof. Let x, y lie in B, and let M, N be two finitely generated Amodules such that xM C!lf and yN C N. Then MN is finitely generated,
and is mapped into itself by multiplication with x ± y and xy.

Corollary. Let A be a ring, K its quotient field, and L a finite separable
extension of K. Let x be an element of L tchich is integral over A. Then
the norm and trace of x from L to K are integral over A, and so are the
coefficients of the irreducible polynomial satisfied by x over K.
Proof. For each isomorphism u of Lover K, ux is integral over A.
Since the norm is the product of ux over all such u, and the trace is the
sum of ux over all such u, it follows that they are integral over A. Similarly, the coefficients of the irreducible polynomial are obtained from the
elementary symmetrie functions of the UX, and are therefore integral
over A.

A ring A is said to be integrally closed in a field L if every element
of L which is integral over A in fact lies in A. It is said to be
integrally closed if it is integrally closed in its quotient field.
Proposition 6. Let A be a N oetherian ring, integrally closed. Let L be
a finite separable extension of its quotient field K. Then the integral closure
of A in L is finitely generated over A.
Proof. It will suffice to show that the integral closure of A is contained
in a finitely generated A.-module, because A is assumed to be ~ oetherian.
Let /t'b . . • ,U'" be a linear basis of Lover K. After multiplying each
tCi by a suitable element of 1'1, we may assurne without loss of generality
that the ICi are integral over 1'1 (Proposition 1). The trace Tl' from L to
K is a K-linear map of L illto K, and is Ilon-degenerate (i.e. there exists
an element x E L sueh that Tr(;l;) ~ 0). If a is a non-zero element of L,
thell the function Tr(a.r) on L is an element of the dual spaee of L (as
K-veetor spaee), and induces a homomorphism of L into its dual spaee.
Sinee the kernel is trivial, it follows that L is isomorphie to its dual under
the bi linear form
(.l', y)

~

Tr(.ry).


[1, §2]

7

INTEGRAL CLOSURE

Let w~, ... , w~ be the dual basis of

Wl, ••• , W n ,

Tr(w~wj)

=

so that

Bij.

cw:

Let c ~ 0 be an element of A such that
is integral over A. Let z be
in L, integral over A. Then zcw~ is integral over A, and so is Tr(czwi)
for each i. If we write

with coefficients bi E K, then

Tr(czwD

=

cb i ,

and cb i E A because A is integrally closed. Hence z is contained in
AC-lWl

+ ... + Ac-lwn •

Since z was selected arbitrarily in the integral closure of A in L, it follows
that this integral closure is contained in a finitely generated A-module,
and our proof is finished.
Proposition 7. If A is a unique factorization domain, then A is integrally closed.

Proof. Suppose that there exists a quotient alb with a, b E A which is
integral over A, and a prime element p in A which divides b but not a.
We have, for some integer n ~ 1,

whence

Since p divides b, it must divide an, and hence must divide a, contradiction.
Theorem 1. Let A be a principal ideal ring, and L a finite separable
extension of its quotient field, of degree n. Let B be the integral closure of
A in L. Then B is afree module ofrank n over A.

Proof. As a module over A, the integral closure is torsion-free, and by
the general theory of principal ideal rings, any torsion-free finitely generated module is in fact a free module. It is obvious that the rank is

equal to the degree [L: Kl.
Theorem 1 is applied to the ring of ordinary integers Z. A finite extension of the rational numbers Q is called a number field. The integral
closure of Z in a number field K is called the ring of algebraic integers of
that field, and is denoted by 0K.


8

[1, §3]

ALGEBRAIC INTEGERS

Proposition 8. Let A be a subring of a ring B, integral over A. Let S
be a multiplicative sub set of A. Then S-1 B is integral over S-1 A. 1f A
is integrally closed, then S-1 A is integrally closed.
Proof. If x E Band SES, and if M is a finitely generated A-module
such that xM C M, then S-IM is a finitely generated S-IA-module
which is mapped into itself by S-I X , so that S-I X is integral over S-1 A.
As to the second assertion, let x be integral over S-1 A, with x in the
quotient field of A. We have an equation

xn

+ bn -

1

Sn-l

xn -

1

+ ... + bo =
So

0,

bi E A and Si E S. Thus there exists an element SES such that sx is
integral over A, hence lies in A. This proves that x lies in S-1 A.

Corollary. 1f B is the integral closure of A in some field extension L
of the quotient field of A, then S-1 B is the integral closure of S-1 A in L.

§3. Prime ideals
Let p be a prime ideal of a ring A and let S = A - p. If B is a ring
containing A, we denote by B p the ring S-IB.
Let B be a ring containing a ring A. Let p be a prime ideal of A and
~ be a prime ideal of B. We say that ~ lies above p if ~ n A = P and
we then write ~Ip. If that is the case, then the injection

induces an injection of the factor rings
Alp ~ B/~,

and in fact we have a commutative diagram :
B~B/~

i

A

~

i

Alp

the horizontal arrows being the canonical homomorphisms, and the
vertical arrows being inclusions.
If B is integral over A, then B I~ is integral over Alp (by Proposition 4).
Nakayama's Lemma. Let A be a ring, a an ideal contained in all maximal ideals of A, and M a finitely generated A-module. 1f aM = ~M, then
M=O.


[I, §3]

PRIME IDEALS

9

Proof. Induction on the number of generators of M. Say M is generated by Wl, .•. ,Wm • There exists an expression

with ai E

Q.

Hence

If 1 - al is not a unit in A, then it is contained in a maximal ideal ll.
Since al E II by hypothesis, we have a contradiction. Hence 1 - al is

a unit, and dividing by it shows that M can be generated by m - 1 elements, thereby concluding the proof.
Proposition 9. Let A be a ring, II a prime ideal, and Ba ring containing
A and integral over A. Then llB -;tf= B, and there exists a prime ideal '.ß
of B lying above ll.

Proof. We know that B u is integral over Au, and that Au is a local ring
with maximal ideal mu. Since we obviously have

it will suffice to prove our first assertion when A is a local ring. In that
case, if llB = B, then 1 has an expression as a finite linear combination
of elements of B with coefficients in ll,

with ai E II and bi E B. Let B o = A[b 1 , ••• ,bnl. Then llB o = B o and
B o is a finite A-module by Proposition 2. Hence B o = 0, contradiction.
To prove our second assertion, we go back to the original notation, and
note the following commutative diagram:
(all arrows inclusions).
We have just proved that muBu -;tf= B u. Hence muB u is contained in a
maximal ideal im of B u, and im n Au therefore contains mu• Since mu is
maximal, it follows that

Let

'.ß = im n B. Then '.ß is a prime ideal of B, and taking intersections


10

[I, §3]

ALGEBRAIC INTEGERS

with A going both ways around
so that

OUf

=

~,

v be an ideal of B, v ~

O.

diagram shows that 9)1 n A

'.ß n A = p,

as was to be shown.
Remark. Let B be integral over A, and let

Then

vn A

~

O.

To prove this, let b E

with ai E A, and ao

~

v, b ~ O.

Then b satisfies an equation

O. But ao lies in

v n A.

Proposition 10. Let A be a subring of B, and assume B integral over A.
Let '.ß be a prime ideal of B lying over a prime ideal ~ of A. Then Iß is
maximal if and only if pis maximal.
Proof. Assume p maximal in A. Then Alp is a field. We are reduced
to proving that a ring wh ich is integral over a field is a field. If k is a field
and x is integral over k, then it is standard from elementary field theory
that the ring k[x] is itself a field, so x is invertible in the ring. Conversely,
assume that '.ß is maximal in B. Then BI'.ß is a field, which is integral
over the ring Alp. If Alp is not a field, it has a non-zero maximal ideal
m. By Proposition 9, there exists a maximal ideal 9)1 of BIIß lying above
m, contradiction.
When an extension is given explicitly by a generating element, then we
can describe the primes lying above a given prime more explicitly.
Let A be integrally closed in its quotient field K, and let E be a finite extension of K. Let B be the integral closure of A in E. Assume that B = A[a]
for some element a, and let feX) be the irreducible polynomial of a over K.
Let p be a maximal ideal of A. We have a canonical homomorphism

A --; Alp

= A,

which extends to the polynomial ring, namely
g(X)

=

m

L
i~l

CiXi ~

m

L

GiXi

=

g(X),

i~l

where G denotes the residue class mod p of an element c E A.
We contend that there is a natural bijection betu'een the prime ideals '.ß of

B lying above p and the irreducible factoTs P(X) of leX) (having leading


[I, §4]

11

CHINESE REMAINDER THEOREM

coefficient 1). This bijection is such that a prime 'l3 of B lying above II corresponds to P if and only if 'l3 is the kernel of the homomorphism
A[a]-+ Ä[a]
where

a is a root of P.

To see this, let

'l3 lie above ll. Then the eanonieal homomorphism

J whieh is eonjugate to a root of some
irredueible faetor of f. Furthermore two roots of J are eonjugate over Ä
if and only if they are roots of the same irredueible faetor of f. Finally,
let z be a root of P in some algebraic closure of Ä. The map
B -+ B/'l3 sends

a on a root of

g(a)

1-+

g(z)

for g(X) E A[X] is a well-defined map, because if g(a)

=

0 then

g(X) = f(X)h(X)

for some heX) E A[X], whence g(z) = 0 also. Being well-defined, our
map is obviously a homomorphism, and since z is a root of an irreducible
polynomial over Ä, it follows that its kernel is a prime ideal in B, thus
proving our contention.
Remark 1. As usual, the assumption that II is maximal can be weakened
to II prime by localizing.
Remark 2. In dealing with extensions of number fields, the assumption
B = A[a] is not always satisfied, but it is true that B~ = A~[a] for all but
a finite number of ll, so that the previous discussion holds almost always

locally. Cf. Proposition 16 of Chapter III, §3.

§4. Chinese remainder theorem
Chinese Remainder Theorem. Let A be a ring, and al, ... , an ideals
such that ai
aj = A for all i ~ j. Given elements Xl, ... ,Xn E A, there
exists X E A such that X == Xi (mod ai) for alt i.

+

Proof. If n

= 2, we have an expression

+

Xla2.
for some elements ai E ai, and we let X = X2al
For each i we can find elements ai E al and bi E ai such that
ai

+ bi =

1,

i !!i; 2.


12

[1, §5]

ALGEBRAIC INTEGERS

The product

n

n

;=2

;=2

TI (ai + bi) is equal to 1, and lies in al + TI ai.
n

al
By the theorem für n

=

+ i=2
TI ai =

Hence

A.

2, we can find an element Yl E A such that
Yl

== 1 (mod al)

Yl

== 0 (mod

Ir ai) .

• =2

We find similarly elements Y2, ... , Yn such that
Yj

Then x =

XlYl

== 0 (mod ai), i

~

j.

+ ... + XnYn satisfies our requirements.

In the same vein as abüve, we observe that if al, ... , an are ideals of
a ring A such that

al
and if 111,

.•• , IIn

+ ... + an =

A,

are positive integers, then
a~l

+ ... + a~n =

A.

The proof is trivial, and is left as an exercise.

§5. Galois extensions
Proposition 11. Let A be a ring, integrally closed in its quotient field K.
Let L be a finite Galois extension of K with group G. Let p be a maximal
ideal of A, and let $, D be prime ideals of the integral closure of A in L
lying above p. Then there exists u E G such that u$ = D.

Proof. Suppose that $

~

uD for any u E G. There exists an element

xE B such that

x == 0 (mod $)
x == 1 (mod uD),

all u E G

(use the Chinese remainder theorem). The norm

NJ«x) =
lies in B n K

TI

uEG

UX

= A (because A is integrally closed), and lies in $ n A = p.


[1, §5]

GALOIS

EXTE~SIO!'i"S

13

But x G!: uO for aH u E G, so that ux G!: 0 for aH u E G. This eontradiets
the fact that the norm of x lies in p = 0 n A.

If one loealizes, one ean eliminate the hypothesis that p is maximal;
just assume that p is prime.

Corollary. Let A be a ring, integrally closed in its quotient jield K.
Let E be a jinite separable extension 01 K, and B the integral closure 01 A
in E. Let p be a maximal ideal 01 A. Then there exists only ajinite number
01 prime ideals 01 B lying above p.

Prool. Let L be the smallest Galois extension of K eontaining E. If

(h, O 2 are two distinet prime ideals uf B lying above p, and ~b ~2 are
two prime ideals of the integral closure of A in L lying above (h and O 2

respeetively, then ~1 ~ ~2' This argument reduees our assertion to the
ease that E is Galois over K, and it then beeomes an immediate eonsequenee of the proposition.
Let A be integrally closed in its quotient field K, and let B be its integral
closure in a finite Galois extension L, with group G. Then uB = B for
every u E G. Let p be a maximal ideal of A, and ~ a maximal ideal of B
lying above p. We denote by G'I.l the subgroup of G eonsisting of those
automorphisms such that O'"~ =~. Then G'I.l operates in a natural way
on the residue class field B/~, and leaves Alp fixed. To eaeh 0'" E G'I.l we
ean assoeiate an automorphism Ü of B/~ over Alp, and the map given by

induees a homomorphism of GIlJ into the group of automorphisms of B/~
over Alp.
The group G'I.l will be ealled the decOInposition group of~. Its fixed
field will be denoted by L d , and will be ealled the decomposition field
of~. Let B d be the integral closure of A in L d , and let 0 = ~ n B d •
By Proposition 11, we know that ~ is the only prime of B lying above O.
Let G = UUjG'I.l be a eoset deeomposition of G'I.l in G. Then the prime
ideals 0'" j~ are preeisely the distinet primes of B lying above p. Indeed,
for two elements 0'", TE G we have O'"~ = T~ if and only if T-IU~ = ~,
i.e. T-1U lies in G'I.l' Thus T, 0'" lie in the same eoset mod G'I.l'
I t is then immediately elear that the deeomposition group of a prime
u~ iSO'"G'I.lO'"-l.

Proposition 12. The jield L d is the smallest subfield E 01 L containing
K such that ~ is the only prime 01 B lying above ~ nE (u'hich is prime in

B nE).


14

[I, §5]

.\LGEBIL\lC I:\TEGEHS

Proof. Let E be us ubove, und let H be the Gulois group of Lover E.
Let q = $ nE. By Proposition 11, aB primes of B lying above q are
conjugate by elements of H. Since there is only one prime, namely $,
it means that H leaves $ invariant. Hence H C G'1l und E =:J L d • We
have already observed thut L d hus the required property.
Proposition 13. Notation being as abol'e, lce haue Alp
ihe canonical injeciion Alp ~ B dle).

=

Bdle (umler

Proof. If (J' is an element of G, not in Gtl, then (J''l.~ ,c. $ and (J'-1i.j3 ,c. $.
Let
Then Cu ~ C. Let x be an element of B". There exists an element y
of B d such that
y ==;1'

(mod C)

y == 1

(mod

,co;

for each (J' in G, but not in Gtl. Hence in particular,

==;r (mod 'l.~)
y == 1 (mod (J'-l'l.~)

y

for each (J' not in G'1l' Thi,.; >,econd congruence yield,.;
(J'y

== 1 (mod

for aB (J' tl Gtl. The norm of y from L" to
factor~ (J'y with (J' tl Gtl. Thus we obtain
N~d(y)==.r

'l.~)

1(

i,. a product of y and other

(mod'l.~).

But the norm lies in 1(, and even in A, since it is a product of elements

integral over A. This last congruence holds mod e, since both ;(; and the
norm lie in B d . This is precisely the meaning of the assertion in our
proposition.
If .r is an element of B, "·e shall denote b~' J it,.; image under the homomorphism B ~ BI'l.~. Then iJ is the automorphism of Bli.j3 satisfying the
relation
iJJ = (J'.r.
If feX) is a polynomial with coefficients in B, we dcnote by fCX) its natural
image under the above homomorphism. Thus, if

feX) = b"X" -+- ... -+- bo.