Elementary and Analytic Theory of Algebraic Numbers

712 Pages·2004·15.33 MB·English

by  Władysław Narkiewicz (auth.)

Springer Monographs in Mathematics
Springer-Verlag Berlin Heidelberg GmbH
Wladyslaw Narkiewicz
Elementary
and Analytic Theory
of Algebraic
Numbers
Third Edition
i
Springer
Wladyslaw Narkiewicz
Wrodaw University
Institute of Mathematics
Pl. Grunwaldzki 2/4
50-384 Wrodaw
Poland
e-mail: [email protected]
Third, revised and extended edition based on the second edition (English):
© PWN-Polish Scientific Publishers, Warszawa 1990
Mathematics Subject Classification (2ooo ):
nRxx, nSxx
ISSN 1439-7382
ISBN 978-3-642-06010-6 ISBN 978-3-662-07001-7 (eBook)
DOI 10.1007/978-3-662-07001-7
Library of Congress Control Number: 2004105720
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2004
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To my wife
Preface
The aim of this book is to present an exposition of the theory of alge
braic numbers, excluding class-field theory and its consequences. There are
many ways to develop this subject; the latest trend is to neglect the classical
Dedekind theory of ideals in favour of local methods. However, for numeri
cal computations, necessary for applications of algebraic numbers to other
areas of number theory, the old approach seems more suitable, although its
exposition is obviously longer. On the other hand the local approach is more
powerful for analytical purposes, as demonstrated in Tate's thesis. Thus the
author has tried to reconcile the two approaches, presenting a self-contained
exposition of the classical standpoint in the first four chapters, and then
turning to local methods.
In the first chapter we present the necessary tools from the theory of
Dedekind domains and valuation theory, including the structure of finitely
generated modules over Dedekind domains. In Chapters 2, 3 and 4 the clas
sical theory of algebraic numbers is developed. Chapter 5 contains the fun
damental notions of the theory of p-adic fields, and Chapter 6 brings their
applications to the study of algebraic number fields. We include here Shafare
vich's proof of the Kronecker-Weber theorem, and also the main properties
of adeles and ideles.
In Chapter 7 we apply analytical methods, and derive functional equations
for various zeta-functions, including Dedekind zeta-functions and Dirichlet's
£-functions. These functions are then applied to the study of asymptotic
distributions of ideals and prime ideals. In Chapter 8 we consider Abelian
extensions of the rationals. We prove the Siegel-Brauer theorem in this case,
obtain the class-number formula, and give an effective bound for negative
quadratic discriminants with class-number one. The last chapter deals with
factorization of algebraic integers into irreducibles.
Each chapter ends with a section containing comments and a short review
of the relevant literature. A short selection of exercises is also given.
At the end of the book we present a choice of open problems containing
some classical questions, and also some problems of more recent vintage.
In the first edition this list contained 35 problems, 14 were added in the
second edition, and now we added 10 more. Some of them were solved in the
meantime. They are marked by an asterisk in our list.
VIII Preface
We expect the reader to have an elementary knowledge of algebraic and
topological notions, including elements of Galois theory.
There are three appendices dealing with locally compact Abelian groups,
Dirichlet series and Baker's method, presenting results utilized in the main
text.
The comments at the end of each chapter have been rewritten to take
account of the development of the subject up to 2003, and the bibliography
has been extended accordingly. To keep the size of the book reasonable some
changes in the main text have been made in comparison to the previous edi
tions. Certain proofs were simplified, and we decided to omit a few theorems.
In contrast to the previous editions, written on a typewriter, this time a com
puter was used, and this gave the possibility to improve the text on several
places. I am grateful to several friends and colleagues, who commented on the
previous editions. A particular thank goes to Dr.Tadeusz Pezda, who carefully
read the outprints of the new version, and suggested several clarifications and
improvements.
Finally I would like to thank the Springer Verlag for the cooperation in
the realization of this book.
Wrodaw, January 2004 Wladyslaw Narkiewicz
Table of Contents
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
1. Dedekind Domains and Valuations . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Valuations and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3. Finitely Generated Modules over Dedekind Domains . . . . . . 24
1.4. Notes to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2. Algebraic Numbers and Integers . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1. Distribution of Integers in the Complex Plane . . . . . . . . . . . . 43
2.2. Discriminants and Integral Bases . . . . . . . . . . . . . . . . . . . . . . . 52
2.3. Applications of Minkowski'e Convex Body Theorem . . . . . . . 66
2.4. Notes to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3. Units and Ideal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1. Valuations of Algebraic Number Fields . . . . . . . . . . . . . . . . . . 85
3.2. Ideal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3. Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.4. Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.5. Notes to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.1. The Homomorphisms of Injection and Norm . . . . . . . . . . . . . 135
4.2. Different and Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.3. Factorization of Prime Ideals in Extensions. More about
the Class Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.4. Notes to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
X Table of Contents
5. l,l}-adic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.1. Principal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.2. Extensions of p-adic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.3. Harmonic Analysis in p-adic Fields . . . . . . . . . . . . . . . . . . . . . . 237
5.4. Notes to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
6. Applications of the Theory of 1-lJ-adic Fields . . . . . . . . . . . . . 257
6.1 Arithmetical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.2. Adeles and Ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
6.3. Notes to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7. Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
7.1. The Classical Zeta-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
7.2. Asymptotic Distribution of Ideals and Prime Ideals . . . . . . . 343
7.3. Chebotarev's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.4. Notes to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
8. Abelian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
8.1. Main Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
8.2. The Class-number Formula and the Siegel-Brauer Theorem 423
8.3. Class-number of Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . 436
8.4. Notes to Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
9. Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
9.1. Elementary Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
9.2. Quantitative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
9.3. Notes to Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · 509
Appendix I. Locally Compact Abelian Groups .............. 511
Appendix II. Function Theory 525
Appendix III. Baker's Method 527
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
Author Index 685
Subject Index 701
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707