Number Theory 2: Algebraic Number Theory

137 Pages·1992·14.46 MB·English

by  A. N. Parshin

A.N. Parshin I.R. Shafarevich (Eds.)
Number Theory II
Algebraic Number Theory
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List of Editors and Author
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki, Editor-in-Chief
Fundamental’nye napravleniya, Vol. 62, Teoriya chisel 2
Publisher VINITI, Moscow 1990 R.V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical
Institute, ul. Vavilova 42, 117966 Moscow, Institute for Scientific Infor-
mation (VINITI), ul. Usievicha 20a, 125219 Moscow, Russia
Consulting Editors
A.N. Parshin, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
I.R. Shafarevich, Steklov Mathematical Institute, ul. Vavilova 42,
117966 Moscow, Russia
Author
Mathematics Subject Classification (1991):
llRxx, 1lSxx H. Koch, Max-Planck-Gesellschaft zur Fijrderung der Wissenschaften e.V.,
Arbeitsgruppe “Algebraische Geometrie und Zahlentheorie” an der
Humboldt-Universitat zu Berlin, Mohrenstr. 39, 1086 Berlin, FRG
ISBN 3-540-53386-9 Springer-Verlag Berlin Heidelberg New York
ISBN O-387-53386-9 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data
Teorila chisel 2. English
Number theory II:
algebraic number theory/A.N. Parshin, I.R. Shafarevich (eds.).
p. cm.+Encyclopaedia of mathematical sciences; v. 62)
Translation of: TeoriG chisel 2, issued as v. 62 of the serial: Itogi nauki i tekhniki.
Serila Sovremennye problemy matematiki. Fundamental’nye napravleniH.
Includes bibliographical references and index.
ISBN 3-540-53386-9 (Berlin).-ISBN O-387-53386-9 (New York)
Algebraic number theory. I. Parshin, A.N. II. Shafarevich, LR. (Igor’ Rostislavovich), 1923-
111. Title. IV. Title: Number theory 2. V. Title: Number theory two. VI. Series.
QA247.T45513 1992 512’.74-dc20 92-2200
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Algebraic Number Fields
H. Koch
Contents
Preface ....................................................... 7
Chapter 1. Basic Number Theory ................................ 8
0 1. Orders in Algebraic Number Fields ........................... 9
1.1. Modules and Orders ................................... 10
1.2. Module Classes ....................................... 12
1.3. The Unit Group of an Order ............................ 15
1.4. The Unit Group of a Real-Quadratic Number Field ........ 17
1.5. Integral Representations of Rational Numbers by Complete
Forms ............................................... 18
1.6. Binary Quadratic Forms and Complete Modules in Quadratic
Number Fields ........................................ 19
1.7. Representatives for Module Classes in Quadratic
Number Fields ........................................ 21
9 2. Rings with Divisor Theory .................................. 22
2.1. Unique Factorization in Prime Elements .................. 22
2.2. The Concept of a Domain with Divisor Theory ............ 23
2.3. Divisor Theory for the Maximal Order of an Algebraic
NumberField ........................................ 25
Q3 . Dedekind Rings ........................................... 27
3.1. Definition of Dedekind Rings ........................... 28
3.2. Congruences ......................................... 29
3.3. Semilocalization ...................................... 30
3.4. Extensions of Dedekind Rings ........................... 30
3.5. Different and Discriminant ............................. 33
3.6. Inessential Discriminant Divisors ........................ 36
3.7. Normal Extensions .................................... 36
3.8. Ideals in Algebraic Number Fields ....................... 40
3.9. Cyclotomic Fields ..................................... 41
3.10. Application to Fermat’s Last Theorem I .................. 43
2 Contents Contents 3
54. Valuations ................................................ 45 1.11. The Principal Ideal Theorem ............................ 103
4.1. Definition and First Properties of Valuations .............. 45 1.12. Local-Global Relations ................................ 104
4.2. Completion of a Field with Respect to a Valuation ......... 49 1.13. The Zeta Function of an Abelian Extension ............... 105
4.3. Complete Fields with Discrete Valuation .................. 49 6 2. Complex Multiplication ..................................... 107
4.4. The Multiplicative Structure of a p-adic Number Field ...... 51 2.1. The Main Polynomial ................................. 107
4.5. Extension of Valuations ................................ 53 2.2. The First Main Theorem ............................... 108
4.6. Finite Extensions of p-adic Number Fields ................ 55 2.3. The Reciprocity Law .................................. 109
4.7. Kummer Extensions ................................... 58 2.4. The Construction of the Ray Class Field .................. 109
4.8. Analytic Functions in Complete Non-Archimedean Valued 2.5. Algebraic Theory of Complex Multiplication .............. 111
Fields ............................................... 59 2.6. Generalization ........................................ 112
4.9. The Elementary Functions in p-adic Analysis .............. 60 0 3. Cohomology of Groups ..................................... 112
4.10. Lubin-Tate Extensions ................................. 62 3.1. Definition of Cohomology Groups ....................... 112
9 5. Harmonic Analysis on Local and Global Fields ................. 63 3.2. Functoriality and the Long Exact Sequence ............... 113
5.1. Harmonic Analysis on Local Fields, the Additive Group .... 64 3.3. Dimension Shifting .................................... 114
5.2. Harmonic Analysis on Local Fields, the Multiplicative Group 65 3.4. Shapiro’s Lemma ..................................... 115
5.3. Adeles ............................................... 66 3.5. Corestriction ......................................... 115
5.4. Ideles ............................................... 68 3.6. The Transgression and the Hochschild-Serre-Sequence ...... 116
5.5. Subgroups of J(K)/K” of Finite Index and the Ray Class 3.7. CupProduct ......................................... 117
Groups .............................................. 69 3.8. Modified Cohomology for Finite Groups ................. 119
3 6. Hecke L-Series and the Distribution of Prime Ideals ............. 70 3.9. Cohomology for Cyclic Groups ......................... 120
6.1. The Local Zeta Functions .............................. 74 3.10. The Theorem of Tate .................................. 121
6.2. The Global Functional Equation ........................ 76 $4. Proof of the Main Theorems of Class Field Theory .............. 121
6.3. Hecke Characters ..................................... 77 4.1. Application of the Theorem of Tate to Class Field Theory ... 121
6.4. The Functional Equation for Hecke L-Series .............. 79 4.2. Class Formations ..................................... 122
6.5. Gaussian Sums ....................................... 80 4.3. Cohomology of Local Fields ............................ 124
6.6. Asymptotical Distribution of Ideals and Prime Ideals ....... 82 4.4. Cohomology of Ideles and Idele Classes .................. 125
6.7. Chebotarev’s Density Theorem .......................... 84 4.5. Analytical Proof of the Second Inequality ................. 129
6.8. Kronecker Densities and the Theorem of Bauer ............ 85 4.6. The Canonical Class for Global Extensions ............... 130
6.9. The Prime Ideal Theorem with Remainder Term ........... 87 Q 5. Simple Algebras ........................................... 131
6.10. Explicit Formulas ..................................... 87 5.1. Simple Algebras over Arbitrary Fields .................... 131
6.11. Discriminant Estimation ............................... 88 5.2. The Reduced Trace and Norm .......................... 132
5.3. Splitting Fields ....................................... 133
5.4. The Brauer Group .................................... 133
Chapter 2. Class Field Theory ................................... 90
5.5. Simple Algebras over Local Fields ....................... 134
,§ 1. The Main Theorems of Class Field Theory ..................... 92 5.6. The Structure of the Brauer Group of an Algebraic Number
1.1. Class Field Theory for Abelian Extensions of Q ............ 92 Field ................................................ 135
1.2. The Hilbert Class Field ................................ 93 5.7. Simple Algebras over Algebraic Number Fields ............ 136
1.3. Local Class Field Theory ............................... 93 9 6. Explicit Reciprocity Laws and Symbols ........................ 137
1.4. The Idele Class Group of a Normal Extension ............. 95 6.1. The Explicit Reciprocity Law of Shafarevich ............... 138
1.5. Global Class Field Theory .............................. 96 6.2. The Explicit Reciprocity Law of Bruckner and Vostokov .... 139
1.6. The Functorial Behavior of the Norm Symbol ............. 97 6.3. Application to Fermat’s Last Theorem II ................. 141
1.7. Artin’s General Reciprocity Law ......................... 98 6.4. Symbols ............................................. 142
1.8. The Power Residue Symbol ............................. 99 6.5. Symbols of p-adic Number Fields ........................ I43
1.9. The Hilbert Norm Symbol .............................. 101 6.6. Tame and Wild Symbols ............................... 144
1.10. The Reciprocity Law for the Power Residue Symbol . . . . . . . . 102 6.7. Remarks about Milnor’s K-Theory ...................... 144
Contents 5
4 Contents
5 7. Further Results of Class Field Theory ............ . . . . 145 Chapter 4. Abelian Fields ....................................... 192
7.1. The Theorem of Shafarevich-Weil ........... . . . . . . 145 3 1. The Integers of an Abelian Field .............................. 193
7.2. Universal Norms ......................... . . . . . . 145 1.1. The Coordinates ...................................... 193
7.3. On the Structure of the Ideal Class Group ... . . . . . . 146 1.2. The Galois Module Structure of the Ring of Integers of an
7.4. Leopoldt’s Spiegelungssatz ................ . . . . . . . . 147 Abelian Field ......................................... 194
7.5. The Cohomology of the Multiplicative Group . . . . 149 4 2. The Arithmetical Class Number Formula ...................... 195
2.1. The Arithmetical Class Number Formula for Complex
Chapter 3. Galois Groups ...................................... 150
Abelian Fields ........................................ 195
0 1. Cohomology of Prolinite Groups ............................. 151 2.2. The Arithmetical Class Number Formula for Real
1.1. Inverse Limits of Groups and Rings ...................... 151 Quadratic Fields ...................................... 197
1.2. Prolinite Groups ...................................... 153 2.3. The Arithmetical Class Number Formula for Real Abelian
1.3. Supernatural Numbers ................................. 154 Fields ............................................... 198
1.4. Pro-p-Groups and p-Sylow Groups ...................... 154 2.4. The Stickelberger Ideal ................................. 200
1.5. Free Prolinite, Free Prosolvable, and Free Pro-p-Groups .... 154 2.5. On the p-Component of the Class Group of Q(cpm) ......... 203
1.6. Discrete Modules ..................................... 155 2.6. Application to Fermat’s Last Theorem III ................. 205
1.7. Inductive Limits in C .................................. 156 Q 3. Iwasawa’s Theory of r-Extensions ............................ 206
1.8. Galois Theory of Infinite Algebraic Extensions ............. 157 3.1. Class Field Theory of r-Extensions ...................... 206
1.9. Cohomology of Prolinite Groups ........................ 159 3.2. The Structure of /i-Modules ............................ 207
1.10. Cohomological Dimension ............................. 159 3.3. The p-Class Group of a r-Extension ..................... 208
1.11. The Dualizing Module ................................. 160 3.4. Iwasawa’s Theorem ................................... 209
1.12. Cohomology of Pro-p-Groups .......................... 161 g 4. p-adic L-Functions ......................................... 211
1.13. Presentation of Pro-p-Groups by Means of Generators and 4.1. The Hurwitz Zeta Function ............................. 212
Relations ............................................ 162 4.2. p-adic L-Functions .................................... 213
1.14. Poincare Groups ...................................... 164 4.3. Congruences for Bernoulli Numbers ..................... 214
1.15. The Structure of the Relations and the Cup Product ........ 165 4.4. Generalization to Totally Real Number Fields ............. 215
1.16. Group Rings and the Theorem of Golod-Shafarevich ....... 166 4.5. The p-adic Class Number Formula ....................... 215
6 2. Galois Cohomology of Local and Global Fields ................ 168 4.6. Iwasawa’s Construction of p-adic L-Functions ............. 216
2.1. Examples of Galois Cohomology of Arbitrary Fields ........ 168 4.7. The Main Conjecture .................................. 218
2.2. The Algebraic Closure of a Local Field ................... 169
2.3. The Maximal p-Extension of a Local Field ................ 171 Chapter 5. Artin L-Functions and Galois Module Structure .......... 219
2.4. The Galois Group of a Local Field ....................... 173
2.5. The Maximal Algebraic Extension with Given Ramification . . 175 Q 1. Artin L-Functions ......................................... 222
2.6. The Maximal p-Extension with Given Ramification ......... 177 1.1. Representations of Finite Groups ....................... 222
2.7. The Class Field Tower Problem ......................... 180 1.2. Artin L-Functions .................................... 224
2.8. Discriminant Estimation from Above ..................... 181 1.3. Cyclotomic Fields with Class Number 1 ................. 226
2.9. Characterization of an Algebraic Number Field by its 1.4. Imaginary-Quadratic Fields with Small Class Number ..... 227
Galois Group ........................................ 182 1.5. The Artin Representation and the Artin Conductor ........ 228
9 3. Extensions with Given Galois Groups ......................... 182 1.6. The Functional Equation for Artin L-Functions .......... 230
3.1. Embedding Problems .................................. 183 1.7. The Conjectures of Stark about Artin L-Functions at s = 0 . 231
3.2. Embedding Problems for Local and Global Fields .......... 185 0 2. Galois Module Structure and Artin Root Numbers ............. 234
3.3. Extensions with Prescribed Galois Group of l-Power Order . . 186 2.1. The Class Group of Z[G] ............................. 235
3.4. Extensions with Prescribed Solvable Galois Group ......... 188 2.2. The Galois Module Structure of Tame Extensions ......... 236
3.5. Extensions with Prescribed Local Behavior ................ 188 2.3. Further Results on Galois Module Structure ............. 236
3.6. Realization of Extensions with Prescribed Galois Group by
Means of Hilbert’s Irreducibility Theorem ................. 190
6 Contents
Appendix 1. Fields, Domains, and Complexes ..................... 237 Preface
1.1. Finite Field Extensions ................................. 237
1.2. Galois Theory ........................................ 238
1.3. Domains ............................................. 238 The purpose of the present article is the description of the main structures and
1.4. Complexes ........................................... 239 results of algebraic number theory. The main subjects of interest are the algebra-
ic number fields. We included in this article mostly properties of general charac-
Appendix 2. Quadratic Residues ................................. 240 ter of algebraic number fields and of structures related to them. Special results
appear only as examples which illustrate general features of the theory.
Appendix 3. Locally Compact Groups . . . . . . . . . . . . 241 On the other hand important parts of algebraic number theory such as class
field theory have their analogies for other types of fields like functions fields of
3.1. Locally Compact Abelian Groups . . . . . . . 241
one variable over finite fields. More general class field theory is part of a theory
3.2. Restricted Products . . . . . . . . . . . . . . . 243
of fields of finite dimension, which is now in development. We mention these
analogies and generalizations only in a few cases. From the application of
Appendix 4. Bernoulli Numbers . . . . . . . . . . . . . . . 243
algebraic number theory to problems about Diophantine equations we explain
the application to the theory of complete forms, in particular binary quadratic
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
forms (Chap. 1.1%7), and to Fermat’s last theorem (Chap. 1.3.10, Chap. 2.6.4,
Chap. 4.2.6).
References . . . . . . . . . . . . . . . . . . . . . 251
A part of algebraic number theory serves as basis science for other parts of
mathematics such as arithmetic algebraic geometry and the theory of modular
AuthorIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
forms. This concerns our Chap. 1, Basic number theory, Chap. 2, Class field
theory, and parts of Chap. 3, Galois cohomology. These domains are presented
Subject Index . . . . . . . . . . . . . . . . . . . . . . . 266
in more detail than other parts of the theory.
In accordance with the principles of this encyclopedia we give in general no
reference to the history of results and refer only to the final or most convenient
source for the result in question. Many paragraphs and sections contain after the
title a main reference which served as basis for the presentation of the material
of the paragraph or section.
Each chapter and some paragraphs contain a detailed introduction to their
subject. Therefore we give here only some technical remarks: At the end of
Theorems, Propositions, Lemmas, Proofs, or Exercises stands the sign 0 or IX~.
The sign 0 means that the Theorem, Proposition, Lemma, or Exercise is easy to
prove or that we gave a full sketch of the proof. The sign in means that one needs
for the proof ideas which are not or only partially explained in this article.
We use Bourbaki’s standard notation such as Z for the ring of integers, Q for
the field of rational numbers, [wf or the field of real numbers, and C for the field
of complex numbers. For hny ring n with unit element, Ax denotes the group of
units, i.e. the group of invertible elements and /i* denotes the set of elements
distinct from 0. Furthermore pL, denotes the group of roots of unity with order
dividing n. For any set M we denote by 1M I the cardinality of M.
I am grateful to I.R. Shafarevich and A.N. Parshin who read a preliminary
version of this book and proposed several important alterations and additions,
to U. Bellack and B. Wi.ist for their help in the production of the final manuscript
and to the staff of Springer-Verlag for their cooperation.
Helmut Koch
0 1. Orders in Algebraic Number Fields 9
its existence for the ring 0, by means of Kronecker’s method. In 0 3 we develop
Chapter 1
systematically the ideal theory of Dedekind rings including the theory of the
Basic Number Theory
different and discriminant and the theory of ramification groups. Nowadays
algebraic number fields are mostly studied by a mixture of ideal and valuation
theory. The latter, being the third method mentioned above, is presented in 4 4.
The first goal of algebraic number theory is the generalization of the theorem p-adic analysis, which is an essential part of the valuation theoretic method, is
on the unique representation of natural numbers as products of prime numbers developed in this article only so far as it is needed in the following. 5 5 explains
to algebraic numbers. Gauss considered the ring Z[fl] of all numbers of harmonic analysis in local and global fields and $6 is devoted to the study of
the form a + fib with a, b E Z and showed that Z[fl] is a ring with L-series and their application to the arithmetic of algebraic number fields.
unique factorization in prime elements (see 9 2.1). He introduced these numbers Chapter 1 of this article covers the content of most of the numerous books on
for the development of his theory of biquadratic residues. Another motivation algebraic number theory which serve as an introduction to the field. We mention
for the study of the arithmetic of algebraic numbers comes from the theory of , only the following: Artin (1951), Borevich, Shafarevich (1985), Eichler (1963)
Tiophantine equations. For example, the quadratic form f(x,, x2) = x1 - Dxi Hasse (1979), Hecke (1923) Lang (1970), Narkiewicz (1990), Weiss (1963), Weyl
with D E Z, fi $ Z can be written in the form (x1 - fixz)(xl + fixJ. (1940).
Hence the question about the representation of integers by f(a,, az) with a,, Artin (1951), Eichler (1963), and Hasse (1979) are introductions to the theory
a2 E Z can be reformulated as a question of factorization of algebraic numbers of algebraic numbers and algebraic functions as well. Narkiewicz (1990) contains
of the form a, + $az. These numbers form a module in the field Q($). a voluminous review about results in the elementary and analytical theory of
Beginning with the year 1840 Kummer (1975) considered the ring of numbers algebraic numbers and a complete bibliography of this domain.
of the form a,, + a, [ + ... + aP-,ip-‘, a,, . . . , aP-l E Z, where p is a prime and c
a primitive p-th root of unity, i.e. [” = 1 and [ # 1. Since in general this ring
has not unique factorization in prime numbers, Kummer introduced “ideal
5 1. Orders in Algebraic Number Fields
numbers” and showed the unique factorization in ideal prime numbers. With
this concept he was able to prove Fermat’s last theorem in many new cases using (Main reference: Borevich, Shafarevich (1985), Chap. 2)
the identity
The subject of this article are the algebraic numbers. A complex number a is
p-1
xp - yp = n (x - ('y). called algebraic if it satisfies an equation of the form
i=O
a” + a,cc”-’ + ... + a, = 0
Up to now Kummer’s deep and beautiful results about cyclotomic fields O(c)
serve as paradigmas for research in algebraic number theory (see Chap. 4) but it with a,, . . . . a, E Q.
needed some 30 years until Kronecker and Dedekind found the right generaliza- An algebraic number field K is a finite field extension of Q lying in C. It is
tion of Kummer’s ideal numbers for arbitrary number fields K: One has to define always of the form K = Q(E) with an algebraic number c(. The degree [K : O] of
the notion of integral algebraic numbers. The integral algebraic numbers con- K over Q is called the degree of K.
tained in K form a ring 0, (9 l.l), which is the natural realm for the generaliza- The first goal of algebraic number theory is the extension of the arithmetic
tion of the unique factorization in prime numbers. in Z and Q to algebraic number fields.
. There are three methods to establish arithmetic in D,. Kronecker considers By arithmetic in Z we mean the unique factorization of natural numbers in
polynomials with coefficients in D, (52.3). Dedekind introduces ideals in DK, the product of prime numbers. The arithmetic in Q is given by the arithmetic in
defining so one of the most important notions of algebra, and the third method Z: Any r E Q - (0) h as a uniquely determined representation
due to Zolotarev and Hensel uses what is nowadays called localization (0 4).
r = (- 1)” n p”d’)
An important part of the theory, in particular the Dirichlet unit theorem, is
valid for rings D in K with K = Q(D), 1 E D and D s D,, called orders of K. P
Orders and modules of elements in K appear in connection with decomposible where the product runs over all prime numbers p and v E (0, l}, VP(r) E Z are
forms (9 1.556). Therefore we begin Chap. 1 with the theory of modules and uniquely determined by r.
orders in algebraic number fields (6 1). In 0 2 we define the notion of a ring with If we want to generalize the arithmetic of Q to an algebraic number field K,
divisor theory formulating axiomatically what one awaits from an arithmetic the first question we have to answer is, what will be the right generalization of
theory of rings. For a domain such a divisor theory is always unique. We indicate H? This should be a ring 0 in K with the following properties:
§ 1. Orders in Algebraic Number Fields 11
10 Chapter 1. Basic Number Theory
1. K is the quotient field of D. Example 1. Let K := Q(a) with an algebraic integer a. Then Z[cl] is an order
2. 0 n Q = z. in K. 0
3. The additive group of D is finitely generated.
Let wr, . . . . w, be a basis of the complete module m in K. Then the dis-
A ring in K with these properties is called an order of K. criminant dKiQ(col, . . . , w,) (see App. 1) is independent of the choice of ol, . . . ,
It is easy to see (compare Example 2) that for K # Q there are infinitely many CD,,I.t is called the discriminant of m and is denoted by d(m).
orders of K. But we shall show in 0 1.1 that there is one maximal order DK Let /I r, . . . , j3, be a basis of D(m) and let A E G&(Q) such that
containing all orders of K, an element a of K belongs to DK if and only if there
(0 1, . . . , qJT = AU&, . . , I#‘.
are integers a,, . . . , a, for some s such that
Then the absolute value (det Al of the determinant of A is independent of the
us + ulcF1 + . .. + a, = 0. (1.1) choice of the bases. It is called the norm of m and is denoted by N(m). It is easy
Furthermore we shall see in Q2 that DK is the natural generalization of Z in the to see that
sense that it is possible to develop arithmetic in DK. The elements of 0, are
d(m) = d(D(m))N(m)2 (1.2)
called the integers of K, and any complex number a satisfying an equation of the
form (1.1) is called an integral algebraic number. and
Orders in a number field K arise in a natural way in connection with modules JWm) = &&V(m) forcxEKX. (1.3)
in K: In this paragraph a module m in K means a finitely generated subgroup
of K+. Since the group K+ has no torsion, m is a free Z-module of rank < d(m) is a rational number, distinct from 0. The sign of d(m) is determined by the
[K : Q]. The module m is called complete (or a lattice in K) if its rank is equal to following proposition.
[K : Q-j.
Proposition 1.3 (Brill’s discriminant Theorem). Let rl be the number of real
Modules play an important role in the arithmetic of K and in number theoret-
embeddings and 2r2 the number of complex embeddings of K in C. Then d(m) is
ical questions connected with algebraic number fields (0 1.5-7). Therefore we
positive f and only ij- r2 is even. q (Narkiewicz (1974), p. 59)
start with the study of modules and orders.
Now let 0 be an order in K. Beside Prop. 1.3 one has the following simple
proposition about d(D).
1.1. Modules and Orders. Let m, and m, be modules in K. Then the product
m1m2 is the abelian group generated by the elements plpZ with p1 E m,, pZ E Proposition 1.4 (Stickelberger’s discriminant Theorem). d(D) is an integer with
m2. Obviously m1m2 is finitely generated, i.e. mlmz is a module in K. d(D) = 1 or d(D) = 0 (mod 4). 0 (Narkiewicz (1974) p. 59)
From the theory of finitely generated abelian groups follows
Proposition 1.1. A number a in an algebraic number field K is an algebraic
integer if and only if there exists a module m # (0) in K with crm c m. Proposition 1.5. Let m,, m, be complete modules with the same order D and
m, 2 m2. Then N(m2)/lv(m,) is equal to the index [m, : m2]. Cl
Proof. Let u be an algebraic integer in K. Then we can take for m the module
generated by 1, CI,a 2, . . . , an-l where n := [K : Q]. On the other hand if we have The main result of the present section is
am c mforsomeaEKandpl,..., pL,i s a basis of m, then there are aij E Z with
Theorem 1.6. The set D, of all algebraic integers in K is an order in K, called
the maximal order.
Cr~i = i aijpj for i = 1, . . . , s.
j=l Proof. Let CI~, a2 E DK and let m, := Z[a,], m2 := Z[a2]. Then
Hence det(adij - aij) = 0.
Let m be a complete module in K. Then (a1 f u2)mlm2 E m1m2 and alt12mlm2 E m1m2.
Hence D, is a ring by Prop. 1.1. It remains to show that D, is finitely generated.
D(m) := (CXE K/am c m}
Let m be a complete module contained in D,. If m # D, take a1 F$D K - m, and
is called the order of m. let (m, al) be the module in K generated by m and CI~. Then D((m, aI)) is a proper
divisor of D(m). If (m, al) # DK take a2 E DK - (m, ar) and so forth. After
Proposition 1.2. The order of a complete module m in K is an order in K. For
finitely many steps we end up with an ~1,s uch that (m, al, . . . , CX,)= Dk. Cl
every order D in K there exists a complete module m in K with D = D(m), e.g.
m=D.Cl The following two propositions are easy to prove.
12 Chapter 1. Basic Number Theory Q 1. Orders in Algebraic Number Fields 13
Proposition 1.7 An algebraic number c( is integral if and only if the minimal An algebraic number is called totally positive if all its real conjugates are
polynomial of CI with respect to Q has integral coefficients. 0 positive. The modules m, and m2 are called equivalent in the narrow sense if
Proposition 1.8. Let f(x) = x”’ + tlIxm-’ + ... + a, E DK[x] and let a E K m2 = am1 with a totally positive a E K”.
The set of classes of equivalent modules with order D will be denoted by
with f(a) = 0. Then a is an integer of K. El
CL(D).
Example 2. A quadratic number field K is by definition an algebraic number
Theorem 1.9. CL(D) is a finite set.
field of degree two. There exists a unique square free d E Z such that K = Q(G).
Let By means of Minkowski’s geometry of numbers (Borevich, Shafarevich (1985),
Chap. 2,§ 3) one gets a more precise result:
(1 + $)/2 if d = 1 (mod 4)
CO=
i Jli if d = 2,3 (mod 4). Theorem 1.10. Let K be an algebraic number field with rI real and 2r, complex
conjugates (App. 1.2), n := [K : Q] = rl + 2r,, and let m be a complete module in
Then (1, o> is a basis of D,.
K with order 0. Then there is an a E m with a # 0 and
An arbitrary order in K is of the form DJ := H[fw] with a positive rational
integer f called the conductor of Df. Cl
Example 3. For number fields K of degree > 2 the maximal order DK is not
always of the form Z[aJ. For instance let K = Q(p) with /I3 + p2 - 28 + 8 = 0. Theorem 1.10 implies Theorem 1.9: Taking in account D G a-lm, Prop. 1.5
Then y = (B + /I”)/2 E 0, and 1, /I, y is a basis of the module DK, but there is no and (1.2), (1.3) one finds
a E 0, with D, = Z[aJ (Hasse (1979), Chap. 25.7, Weiss (1963), p. 170). q
[a-‘m : Xl] = N(a-‘m)-’ = I%,&4 G 4 ““!Jid(DII. (1.4)
The discriminant d, := d(DJ is called the discriminant of K. Beside the degree N(m) 0 G nn
[K : Q] the discriminant is the most important invariant of an algebraic number
field K. By the discriminant theorem of Hermite (Theorem 1.12) there are only Hence in each class of CL(D) we have a module m’ containing D such that
finitely many algebraic number fields with given discriminant. Hence one can [m’ : D] is limited. It follows from the theory of finitely generated abelian groups
classify algebraic number fields by means of their degree and their discriminant, that there are only finitely many such modules in CL(D). Cl
of course conjugate fields have the same discriminant. By Example 2 the dis-
For the proof of Theorem 1.10 we consider complete modules as lattices in R”
criminant of the quadratic field K = Q(s) is given as follows: d, = d if d E in the following way: Let gl, . . . , gl, be the real isomorphisms of K and let g,, +1,
1 (mod 4), d, = 4d if d z 2,3 (mod 4).
9 ,.+1, . . . , gl, gn be the pairs of conjugate complex isomorphisms of K, r =
This shows that a quadratic field is uniquely determined by its in discriminant.
rl + r2. Then
This is not true in general for fields of higher degree (Example 19). See Table 5
for non-conjugate totally real cubic fields with the same discriminant < 500000. ~:a~(g1a,...rg~,a,Reg,,+la,...,Reg,a,Img,,+la,..., Im g,a)
See $6.11 and Chap. 3.2.8 for discriminant estimation and for the smallest
defines an injection of K in KY’.I t is easy to see that for m E %Jl(D) the image 4(m)
discriminant for fields with given degree.
is a lattice in R” with discriminant 2-rzdm.
We are going to apply the following geometrical theorem to 4(m): Minkow-
1.2. Module Classes. Let 0 be an order in the algebraic number field K.
ski’s convex body theorem. Let L be a lattice in IR” with discriminant 1. Further-
. Obviously for any a E K” the complete module aD has order D. If all complete
more let M be a convex closed set in R” which is central symmetric with respect
modules with order 0 have the form aD for some a E K”, then D is a principal
to the origin and has volume I/(M) with V(M) 3 2”l. Then M contains at least
ideal ring and there is an arithmetic in D which is quite similar to the arith-
one point of L distinct from the origin. q (Hasse (1979) Chap. 30.2)
metic in Z (see $2.1). In general K acts on the set !VI(O) of complete modules
with order D by multiplication. The orbits of this action are called module The application of Minkowski’s convex body theorem is based on the
classes, i.e. a module class of D consists of all modules of the form am, a E K”, following
for some fixed module m E ‘%X(D).T wo modules lying in the same class are called
Lemma. Let m E W(O) and let M be a convex closed set which is central
equivalent. The number h(Q) of module classes of 0 measures the deviation of
symmetric with respect to the origin and contained in
D from being a principal ideal domain. We show in this section that h(D) is finite.
In the application to the theory of quadratic forms (§ 1.6) we need also a finer
classification of modules:
4 1. Orders in Algebraic Number Fields 15
14 Chapter 1. Basic Number Theory
212 n2n
Then there is a number c1 # 0 in nt with
‘(n!)2
and in particular Id( > 1 for n > 1. Cl
Proof. We apply Minkowski’s convex body theorem to an M and the lattice Furthermore by means of Stirling’s formula we get the estimation
@(m) where t > 0 is determined by
Id( > (a)i’*&exp(2n - &)
V(M) = (2t)VJld(mN.
Then the assumptions of the theorem are fulfilled. Hence there is an element for n > 1, called Minkowski’s discriminant bound (see 5 6.15 and Chap. 3.2.8 for
@(cc) # 0 in t&m) with a E m. Now @(cc) E N implies (1.5). further results on discriminant estimation).
It remains to choose M. We put Minkowski’s convex body theorem is also useful for the proof of the following
discriminant theorem of Hermite:
M = (A,, . . . . UER” 2 IJ-,I+2 i Theorem 1.12. Let N be a natural number. There are only finitely many algebraic
v=l v=*,+l
number fields K with Id( < N. 0 (Narkiewicz (1974), Theorem 2.11)
This is obviously a convex closed set in R” which is central symmetric with
respect to the origin. Since the geometrical mean is smaller or equal the arith-
1.3. The Unit Group of an Order. Let D be an order in the algebraic number
metical mean, we have M 5 N. Now Theorem 1.10 follows from the computa-
field K with rl real and 2r, complex conjugates. The structure of the group XI’
tion of the volume of M:
of invertible element in Q is given by Dirichlet’s unit theorem:
2*I-9fZn”
Theorem 1.13. nx is the direct product of the finite cyclic group of roots of
V(M)= n, .Cl
unity in D and a free abelian group of rank r1 + r2 - 1.
Example 4. Let K = Q(fi) as in Example 2. Then The proof of Theorem 1.13 proceeds in several steps.
a) Let gl, . . . . g,, be as in the proof of Theorem 1.10 and put 1, = 1 for v = 1,
d if d = 1 (mod 4) . . . . rI and 1, = 2for v = rl + 1, . . . . r. The logarithmical components of c(E K”
WA) =
4d if d G 2,3 (mod 4). are defined as l,(u) := 1, loglg,cll for v = 1, . . . . r. Furthermore we put l(a) :=
According to (1.4) for the computation of ICL(D,)I we have to consider the (4 (co, * . . 3 l,(a)). Then 1 is a homomorphism of Kx into the additive group of KY.
Moreover 0” is mapped into the subspace U of R’ given by
complete modules m’ 2 DK with
u = ((A,, . ..) A”,)E R’ 1 I, + ... + A, = O}.
b) The kernel of the map 1: 0 ’ -+ R’ is finite. This follows from
Lemma 1.14. Let c be a positive real number. Then there are only finitely many
a) Let d < 0 and therefore r, = 1. Then ; ;$m < 2 if Id( < 10. crEBwithlg,al<cforv=l,..., n.
0
Hence ICL(D,)I = 1 ifd = -1, -2, -3, -7. Proof. Let or, . . . , w, be a basis of D and kl, . . , K, the complementary basis
‘b) Let d > 0 and therefore rz = 0. Then +Jm < 2 if Id( < 16. Hence (Appendix 1.1). For CI = h,o, + ... + h,o,, h, E Z, we have
ICL(DK)I = 1 if d = 2, 3, 5. Cl
h, = trK,duK,) = g,ag, IC, + ... + g,,ag,,%
Applying Theorem 1.10 to m = D = DK we find
Ihvl d c(ls~d + ... + lwvl) forv=l,...,n.O
Since the kernel of 1 is a finite group, it is a group of roots of unity, and since all
roots of unity in D are mapped by 1 to 0 the kernel of 1 is the group of all roots
This proves the following discriminant theorem of Minkowski: of unity in 0.
c) From Lemma 1.14 it follows also that l(E)‘) is a discrete subgroup of U,
Theorem 1.11. Let K be an algebraic number field with rl real and 2r, complex and since the vector space U has dimension r - 1 the abelian group l(S)‘) has
conjugates. Then rank < r - 1.