Graduate Texts in Mathematics

11 O

Editorial Board

S. Axler F.W. Gehring

Springer Science+Business Media, LLC

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K.A. Ribet


BOOKS OF RELATED lNTEREST BY SERGE LANG

Linear Algebra, Third Edition
1987, ISBN 96412-6
Undergraduate Algebra, Second Edition
1990, ISBN 97279-X
Complex Analysis, Fourth Edition
1999, ISBN 98592-1
Real and Functional Analysis, Third Edition
1993, ISBN 94001-4
Introduction to Algebraic and Abelian Functions, Second Edition
1982, ISBN 90710-6
Cyclotomic Fields 1 and II
1990, ISBN 96671-4

0THER BOOKS BY LANG PUBLISHED BY
SPRINGER-VERLAG
Introduction to Arakelov Theory • Riemann-Roch Algebra (with William Fulton) •
Complex Multiplication • Introduction to Modular Forms • Modular Units (with Daniel
Kubert) • Fundamentals of Diophantine Geometry • Introduction to Complex Hyperbolic Spaces • Elliptic Functions • Number Theory III • Algebraic Number Theory •
SL,(R) • Abelian Varieties • Differential and Riemannian Manifolds • Undergraduate
Analysis • Elliptic Curves: Diophantine Analysis • lntroduction to Linear Algebra •
Calculus of Severa! Variables • First Course in Calculus • Basic Mathematics •
Geometry: A High School Course (with Gene Murrow) • Math! Encounters with High
School Students • The Beauty of Doing Mathematics • THE FILE

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Serge Lang

Alge braic N um ber
Theory
Second Edition

'Springer

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Serge Lang
Department of Mathematics
Yale University
New Haven, CT 06520
USA

Editorial Board

S. Axler
Mathematics Department
San Francisco State
University
San Francisco, CA 94132
USA

F.W. Gehring
Mathematics Department
East Hali
University of Michigan
Ann Arbor, MI 48109
USA

K.A. Ribet
Mathematics Department
University of California
at Berkeley
Berkeley, CA 94720-3840
USA

Mathematics Subject C!assifications (1991): llRxx, llSxx, llTxx
With 7 Illustrations
Library of Congress Cataloging-in-Publication Data
Lang, Serge, 1927Algebraic number theory f Serge Lang. - 2nd ed.
p. cm.- (Graduate texts in mathematics; 110)
Includes bibliographical references and index.
ISBN 978-1-4612-6922-9

ISBN 978-1-4612-0853-2 (eBook)
DOI 10.1007/978-1-4612-0853-2

1. Algebraic number theory.
QA247.L29 1994
512'.74---dc20

I. Title.

II. Series.
93-50625

Originally published in 1970 © by Addison-Wesley Publishing Company, Inc., Reading,
Massachusetts.

© 1994, 1986 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1986
Softcover reprint of the hardcover 2nd edition 1986
All rights reserved. This wark may not be translated or copied in whole or in part without
the written permission of the publisher Springer Science+Business Media, LLC.
except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval,
electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication,
even if the forrner are not especially identified, is not to be taken as a sign that such names,
as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used
freely by anyone.

9 8 7 6 5 4 3 (Corrected third printing )

ISBN 978-1-4612-6922-9

SPIN 10772455

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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 1 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 W eber, 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, 1 have intermingled the ideal
and idelic approaches without prejudice for either. 1 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 a reader
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

V

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Preface for the Second Edition
The principal change in this new edition is a complete rewriting of
Chapter XVII on the Explicit Formulas. Otherwise, I have made a
few additions, and a number of corrections. The need for them was
pointed out to me by severa! people, but I am especially indebted to
Keith Conrad for the list he provided for me as a result of a very careful
reading of the book.
New Haven, 1994

SERGE LANG

vi

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Prerequi sites
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 some analysis.
Chapter XIV assumes Fourier analysis on locally compact groups. Chapters XV through XVII assume only standard analytical facts (we even

prove some of them), except for one allusion 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 small interval to the left of 1. If one knew this,
the proof would become only a page long, and the L-series 'vould not be
needed at all. W e 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 I and IL
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.
W e use the o and O notation. If /, g are two functions of a real variable,
and g is always ~ O, we write f = O(g) if there exists a constant C > O
such that lf(x)l ~ Cg(x) for all sufficiently large x. We writef = o(g) if
lim"_..,f(x)/g(x ) = O. We writef ~ g if lim"_..,f(x)jg(x ) = 1.

vii

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Conten ts
PartOne

General Basic Theory
I

CHAPTER

Algebraic lntegers

1. Localization .
2. Integral closure
3. Prime ideals .
4. Chinese remainder theorem
5. Galois extensions
6. Dedekind rings
7. Discrete valuation rings
8. Explicit factorization of a prime
9. Projective modules over Dedekind rings

CHAPTER

3

4
8
11

12
18
22
27


29

II

Completion s

1. Definitions and completions
2. Polynomials in complete fields
3. Some filtrations .
4. Unramified extensions
5. Tamely ramified extensions

31
41

45
48
51

CHAPTER

III

The Different and Discriminan t

1.
2.
3.

Complement ary modules

The different and ramification
The discriminant

57

62
64

IX

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CONTENTS

X

CHAPTER

IV

Cyclotomic Fields
1.

2.
3.
4.

Roots of unity
Quadratic fields

Gauss sums
Relations in ideal classes

71

76
82
96
CHAPTER

V

Parallelotopes

1.
2.
3.
4.

The product formula
Lattice points in parallelotopes
A volume computation
Minkowski's constant

CHAPTER

99
110
116
119


VI

The Ideal Function

1.
2.
3.

Generalized ideal classes
Lattice points in homogeneously expanding domains
The number of ideals in a given class .

CHAPTER

123
128
129

VII

ldeles and Adeles

1. Restricted direct products
2. Adeles .
3. Ideles .
4. Generalized ideal class groups; relations with idele classes
in the idele classes .
5. Embedding of
6. 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

1. Lemmas on Dirichlet series .
2. Zeta function of a number field
3. The L-series .
4. Density of primes in arithmetic progressions .
5. Faltings' finiteness theorem

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155
159
162
166

170



CONTENTS

xi

PartTwo
Class Field Theory
CHAPTER

IX

Norm Index Computatio ns

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

Algebraic preliminaries .
Exponential and logarithm functions
The local norm index
A theorem on units .
The global cyclic norm index
Applications .

CHAPTER


179
185
187
190
193
195

X

The Artin Symbo1, Reciprocity Law, and C1ass Fie]d Theory

1. Formalism of the Artin symbol
2. Existence of a conductor for the Artin symbol
3. Class fields

CHAPTER

197
200
206

XI

The Existence Theorem and Local CJass Field Theory
1. Reduction to Kummer extensions
2. Proof of the existence theorem .
3. The complete splitting theorem
4. Local class field theory and the ramification theorem
5. The Hilbert class field and the principal ideal theorem
6. Infinite divisibility of the universal norms


CHAPTER

213
215
217
219
224

226

XII

L-series Again
1.

2.
3.

The proper abelian L-series
Artin (non-abelian) L-series
Induced characters and L-series contributions

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229

232
236



xii

CONTENTS

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

1.

The Poisson summation formula

2. A special computation
3. Functional equation .
4. Application to the Brauer-Siegel theorem
5. Applications to the ideal function
Appendix: Other applications

245
250
253
260
262
273

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 series
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

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322
323
325
327
328


CONTEN'fS
CHAPTER

XIII

XVII

Explicit Formulas

331
333
337

1. Weierstrass factorization of the L-series
2. An estimate for ~' / ~ .
3. The Weil formula
4. The basic sum and the first part of its evaluation
5. Evaluation of the sum: Second part

344
348

Bibliography


353

Index .

355

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PART ONE
BASIC THEORY

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CHAPT ER I

Algeh raic lntege rs
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
commutat ive algebra.

§1. Localiza tion
Let A bea ring. By a multiplic ative 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 throughou t that O does
not lie in the subset.
Let K be the quotient field of A, and let S be a multiplica tive subset
of A. By s- 1 A we shall denote the set of quotients x/8 with X in A and
8 inS. It is a ring, and A has a canonica! inclusion in s- 1 A.
If M is an A-module contained in some field L (containin g K), then
s- 1M denotes the set of elements v/8 with VE M and 8 E s. 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 A as subring.
Let p bea prime ideal of A (by definition, p ~ A). Then the complement of pin A, denoted by A - p, is a multiplica tive subset S = Sp of A,
and we shall denote s- 1 A by Ap.
A local ring is a ring which has a unique maxima! ideal. If o is such a
ring, and m its maxima! ideal, then any element x of o not lying in m
must be a unit, because otherwise, the principal ideal xo would be contained in a maxima! ideal unequal to m. Thus m is the set of non-units
of o.
3

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4

[I, §2]

ALGEBRAIC INTEGERS

The ring A~ defined above is a local ring. As can be verified at once,
its maxima! ideal m~ consists of the quotients x/8, with x in p and 8 in A
but not in p.
We observe that m~ n A = p. The inclusion J is clear. Conversely,

if an element y = x/8lies in m~ n A with x E p and 8 E S, then x = 8Y E p
and 8 f1. p. Hence y E p.
Let A be a ring and S a multiplicative subset. Let a' be an ideal of
s- 1 A. Then
a' = s- 1 (a' n A).
The inclusion J is clear. Conversely, let x Ea'. Write x = a/8 with
some a EA and 8 E s. Then 8X Ea' n A, whence X E s- 1 (a' n A).
Under multiplication by s-I, the multiplicative system of ideals of A
is mapped homomorphically onto the multiplicative system of ideals of
s- 1A. This is another way of stating what we have just proved. If a
1a is the unit ideal, then it is clear that a n S is
is an ideal of A and
not empty, oras we shall also say, a meets S.

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 I. There exi8t8 a jinitely generated non-zero A-module M C L 8uch
that xM CM.
INT 2. The element x 8ati8fie8 an equation

with coejficient8 a; E A, and an integer n ~ 1. (Such an equation
will be called an integral equation.)

The two conditions are actually equivalent. Indeed, assume INT 2.
The module M generated by 1, x, ... , xn- 1 is mapped into itself by the
element x. Conversely, assume there exists M = (v1, ... , vn) such that

xM C M, and M ~ O. Then

with coefficients a;i in A. Transposing xv 1 ,

... ,

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xvn to the right-hand side


5

INTEGRAL CLOSURE

[I, §2]

of these equations, we conclude that the determinant

x-au

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 ;:o!! O of A such that cx is integral over A.
Proof. There exists an equation

with ai EA and an

;:o!!


O. Multiply it by a~- 1 • Then
(anx)n

+ · · · + aoa~-I =

O

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

Proposition 3. Let AC B C C be three rings. lf 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
Xn

+ bn-IXn-l + •· · + bo =

0

with biE B. Let B1 = A[b 0 , ••• , bn_ 1]. Then B 1 is a finitely generated
A-module by Proposition 2, and B 1 [x] is a finitely generated B 1-module,
whence a finitely generated A-module. Since multiplication by x maps
B1[x] into itself, it follows that x is integral over A.
Proposition 4. Let A C B be t'WO rings, and B integral over A. Let u
bea homomorphism of B. Then u(B) is integral over u(A).


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6

ALGEBRAIC INTEGERS

[I, §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 elernents of L 'which 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 M and yN C N. Then !liN 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 La finite separable
extension of K. Let x be an element of L which is integral over A. Then
the norrn and trace of x frorn L to K are integral over A, and so are the
coe.fficients of the irreducible polynornial satisfied by x over K.
Proof. For each isomorphism u of L over 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 symmetric functions of the ux, and are therefore integral
over A.


A ring A is said tobe 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 bea Noetherian ring, integraUy closed. Let L be
a finite separable extension of its quotient field K. Then the integral closure
of A in Lis 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 tobe !\oetherian.
Let w 1 , ••. , tl'n be a linear hasis of L over K. After multiplying each
u·; by a suitable element of A, we may assume without loss of generality
that the te; are integral over A (Proposition 1). The trace Tr from L to
K is a K-linear map of L into K, and is non-degenerate (i.e. there exists
an element x EL such that Tr(:r) ~ 0). If a is a non-zero element of L,
then the function Tr(ax) on L is an element of the dual space of L (as
K-vector space), and induces a homomorphism of L into its dual space.
Since the kernel is trivial, it follows that Lis isomorphic to its dual under
the bilinear form

(.r, y)

~---+

Tr(.ry).

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[I, §2]


7

INTEGRAL CLOSURE

Let wf, ... , w~ be the dual hasis of w1. ... ,
Tr(w~wi)

=

Wn,

so that

Oii·

Let c ~ O be an element of A such that cw~ 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 coeffi.cients bi E K, then
Tr(czwD = cbi,

and cbi E A because A is integrally closed. Hence z is contained in
Ac- 1w1

+ · · · + Ac- wn.
1

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.
Propositi on 7. lf A is a unique factorization domain, then A is integrally closed.

Proof. Suppose that there exists a quotient ajb 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,
(ajb)n

+ an-l(ajb) n-l + · · · + ao =O,

whence

Since p divides b, it must divide an, and hence must divide a, contradict ion.
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 a free module of rank 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: K].

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 ox.

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8

ALGEBRAIC INTEGERS


[I, §3]

Proposition 8. Let A bea 8Ubring of a ring B, integral over A. Let 8
bea multiplicative 8ub8et of A. Then 8- 1B i8 integral over 8- 1A. lf A
i8 integrally clo8ed, then 8- 1 A i8 integrally clo8ed.
Proof. If x E B and 8 E 8, and if M is a finitely generated A-module
such that xM c M, then 8- 1M is a finitely generated 8- 1 A-module
which is mapped into itself by 8- 1x, so that 8- 1x is integral over 8- 1 A.
As to the second assertion, let x be integral over 8- 1 A, with x in the
quotient field of A. We have an equation
xn

+ bn-1 xn-1 + ... + bo = o,
8n-1

So

b, E A and 8i E 8. Thus there exists an element 8 E 8 such that 8X is
integral over A, hence lies in A. This proves that X lies in 8- 1A.
Corollary. lf B i8 the integral clo8ure of A in 8ome field extension L
of the quotient field of A, then 8- 1B i8 the integral closure of 8- 1A in L.

§3. Prime ideals
Let p be a prime ideal of a ring A and let 8 = A - p. If B is a ring
containing A, we denote by B~ the ring 8- 1B.
Let B bea ring containing a ring A. Let ll bea prime ideal of A and
~ bea prime ideal of B. We say that ~ lies above ll if ~ n A = ll and
we then write ~lll· If that is the case, then the injection


induces an injection of the factor rings
A/p ~ B/~,

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

Ỵ

A~

Ỵ

A/p

the horizontal arrows being the canonica! homomorphisms, and the
vertical arrows being inclusions.
If B is integral over A, then B/~ is integral over A/p (by Proposition 4).
Nakayama's Lemma. Let Abea ring, a an ideal contained in all maximal ideal8 of A, and Ma jinitely generated A-module. lf aM = M, then
M=O.

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[I, §3]

PRIME IDEALS

9

Proof. Induction on the number of generators of M. Say M is generated by w 1 , ••. , Wm. There exists an expression


with ai E a. Hence

If 1 - a 1 is not a unit in A, then it is contained in a maxima! idealtJ.

Since a 1 E lJ by hypothesis, we have a contradiction. Hence 1 - a 1 is
a unit, and dividing by it shows that M can be generated by m- 1 elements, thereby concluding the proof.
Proposition 9. Let Abea ring, lJ a prime ideal, and Ba ring containing

A and integral over A. Then !JB
of B lying above lJ.

~

B, and there exists a prime ideal

~

Proof. We know that Bp is integral over Ap, and that Ap is a local ring
with maxima! ideal mp. Since we obviously have

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

with ai E lJ and biE B. Let B 0 = A[bt, ... , bn]. Then tJBo = B 0 and
B 0 is a finite A-module by Proposition 2. Hence B 0 = O, 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 mpBp ~ Bp. Hence mpBp is contained in a
maximal ideal IDl of Bp, and IDl n Ap therefore contains mp. Since mp is
maximal, it follows that

Let

~

= IDl

n B. Then ~ is a prime ideal of B, and taking intersections

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10

[I, §3]

ALGEBRAIC INTEGERS

with A going both ways around our diagram shows that 9Jl n A = p,
so that
'13 n A = p,
as was to be shown.
Remark. Let B be integral over A, and let b be an ideal of B, b
Then b n A~ O.

To prove this, let b E b, b


with ai EA, and a 0

~

~

~

O.

O. Then b satisfies an equation

O. But a 0 lies in b n A.

Proposition 10. Let A bea subring of B, and assume B integral over A.
Let '13 be a prime ideal of B lying over a prime ideal p of A. Then '13 is
maximal if and only if p is maximal.

Proof. Assume p maximal in A. Then A/p is a field. We are reduced
to proving that a ring which 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 '13 is maximal in B. Then B/$ is a field, which is integral
over the ring A/p. If Ajp is not a field, it has a non-zero maximal ideal
m. By Proposition 9, there exists a maximal ideal 9Jl of B/$ 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 f(X) be the irreducible polynomial of a over K.

Let p bea maximal ideal of A. We have a canonical homomorphism
A

---t

Ajp =

A,

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

=

i=l

where

m

m

L: c;Xi

f---+

L: ciXi =

g(X),


i=l

c denotes the residue class mod p of an element c E A.

W e contend that there is a natural bijection between the prime ideals $ of
B lying above p and the irreducible factors P(X) of f(X) (having leading

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[1, §4]

11

CHINESE REMAINDER THEOREM

coe.fficient 1). This bijection is such that a prime 'l3 of B lying above ll corresponds to P if and only if 'l3 is the kernel of the homomorphism

A[a]-t A[a]
where a is a root of P.

To see this, let 'l3 lie above lJ. Then the canonica! homomorphism
sends a on a root of 1 which is conjugate to a root of some
irreducible factor of 1. Furthermore two roots of 1 are conjugate over A
if and only if they are roots of the same irreducible factor of]. Finally,
let z bea root of Pin some algebraic closure of A. The map

B -t B/'l>

g(a)


~

g(z)

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

=

O then

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

for some h(X) E A[X], whence g(z) = O also. Being well-defined, our
map is obviously a homomorphism, and since z is a root of an irreducible
polynomial over A, it follows that its kernel is a prime ideal in B, thus
proving our contention.
Remark 1. As usual, the assumption that lJ is maximal can be weakened
to lJ prime by localizing.
Remark 2. In dealing with extensions of number fields, the assumption
= A[a] is not always satisfied, but it is true that Bp = Ap[a] for all but
a finite number of \), so that the previous discussion holds almost ahvays
locally. Cf. Proposition 16 of Chapter III, §3.
B

§4. Chinese remainder theorem
Chinese Remainder Theorem. Let Abea ring, and a 1 , . . . , an ideals
such that ai+ ai= A for all i ~ j. Given elements x 1 , ••. , Xn EA, there
exists x E A such that x = Xi (mod a;) for all i.
Proof. If n


= 2, we have an expression

+

for some elements a; Ea;, and we let x = x 2 a 1
x 1a 2 •
For each i we can find elements a; E a 1 and biE ai such that

ai+ b; = 1,

i ;;;; 2.

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12

ALGEBRAIC INTEGERS

The product

[I, §5]

n

n
II
(ai+ bi) is equal to 1, and lies in a 1 + II lli.
i=2

i=2

n

lll

By the theorem for n

=

+II
lli =A.
i=2

2, we can find an element y 1 EA such that
Y1

=1

Y1

= O (mod

We find similarly elements y 2 ,

(mod n1)

••• , Yn

fr ai) ·


•=2

such that

Yi =O (mod

Then x = x1y 1

lli),

i

-;6

j.

+ · · · + XnYn satisfies our requirements.

In the same vein as above, we observe that if a1 ,
a ring A such that
ll1

and if 111,

Hence

••• , lin

••• ,


an are ideals of

+ •••+ lln =A,

are positive integers, then

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

§5. Galois extensio ns
Propositi on 11. Let A bea ring, integrally closed in its quotient field K.
Let L bea finite Galois extension of K with group G. Let ll bea maximal
ideat of A, and tet '.13, O be prime ideals of the integrat closure of A in L
tying above )l. Then there exists O' E G such that O''.j3 = O.

Proof. Suppose that '.13
x E B such that

-;6

0'0 for any O'

x =O (mod '.13)
x = 1 (mod 0'0),

E G.

There exists an element

ali O' E G


(use the Chinese remainder theorem). The norm
N§c(x) =

II O'X

uEG

lies in B n K

= A (because A is integrally closed), and lies in '.13 n A = ll·

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[1, §5]

GALOIS

EXTE~SIO~S

13

But x E;t: uO for all u E G, so that ux Et: O for all the fact that the norm of x lies in lJ = O n A.
If one localizes, one can eliminate the hypothesis that lJ is maxima!;
just assume that lJ is prime.
Corollary. Let A be a ring, integrally closed in its quotient field K.
Let E bea finite separable extension of K, and B the integral closure of A
in E. Let lJ be a maximal ideal of A. Then there exists only a finite number

of prime ideals of B lying above p.
Proof. Let L be the smallest Galois extension of K containing E. If
2 are two distinct prime ideals of B lying above lJ, and 1lh, $ 2 are
two prime ideals of the integral closure of A in L lying above 0 1 and 0 2
respectively, then $ 1 r!= $ 2 . This argument reduces our assertion to the
case that E is Galois over K, and it then becomes an immediate consequence of the proposition.

Ot, 0

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 lying above lJ. W e denote by Gautomorphisms such that u$ = $. Then Gon the residue class field B/$, and leaves Ajp fixed. To each u E Gcan associate an automorphism it of B/$ over Ajp, and the map given by

induces a homomorphism of Gover A/p.
The group Gfield will be denoted by La, and will be called the decomposition field
of $. Let Ba be the integral closure of A in La, and let O = $ n Ba.
By Proposition 11, we know that $ is the only prime of B lying above O.
Let G = UuiGideals u i$ are precisely the distinct primes of B lying above p. Indeed,
for two elements u, TE G we have i.e. r- 1u lies in GIt is then immediately clear that the decomposition group of a prime
u$ is uGProposition 12. The field Ld is the smallest subfield E of L containing

K such that $ is the only prime of B lying above $ nE (u·hich is prime in
B nE).

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