An Introduction to Algebraic Number Theory

232 Pages·1990·7.673 MB·English

by  Takashi Ono (auth.)

THE UNIVERSITY SERIES IN MATHEMATICS
Series Editor: Joseph J. Kohn
Princeton University
THE CLASSIFICATION OF FINITE SIMPLE GROUPS
Daniel Gorenstein
VOLUME 1: GROUPS OF NONCHARACTERISTIC 2 TYPE
ELLIPTIC DIFFERENTIAL EQUATIONS AND
OBSTACLE PROBLEMS
Giovanni Maria Troianiello
FINITE SIMPLE GROUPS: An Introduction to Their Classification
Daniel Gorenstein
AN INTRODUCTION TO ALGEBRAIC NUMBER THEORY
Takashi Ono
INTRODUCTION TO PSEUDODIFFERENTIAL
AND FOURIER INTEGRAL OPERAT ORS
Fram;ois Treves
VOLUME 1: PSEUDODIFFERENTIAL OPERATORS
VOLUME 2: FOURIER INTEGRAL OPERATORS
MATRIX THEORY: A Second Course
James M. Ortega
A SCRAPBOOK OF COMPLEX CURVE THEORY
C. Herbert Clemens
TOPICS IN NUMBER THEORY
J. S. Chahal
An Introduction to
Algebraic N utnber Theory
Takashi Ono
The Johns Hopkins University
Baltimore, Maryland
Plenum Press • New York and London
Llbrary of Congress Cataloglng-ln-Publlcatlon Data
Ono, Takash i.
[Süron josetsu. Engl ishl
An lntroduetion to algebrale number theory / Takash1 Ono. -- 2nd
ed.
p. cm. -- (The University series in mathe.aties)
Translation of: Süron josetsu.
1neludes bibliographleal references.
[SNB-[3:978-1-4612-7872-6 e-[SBN-13: 978-1-4613-0573-6
DOI: 10.1007/978-1-4613-0573-6
1. Algebraie number theory. I. Tttle. 11. Series: University
sertes in mathemattes (Plenum Press)
CA247.056 1990
512' .74--de20 90-30155
CIP
This volume is a translation (by the author) of the
Second Edition of Suron Josetsu (An Introduction to
Algebraic Number Theory), originally published by
Shokabo Publishing Co., Ltd., Tokyo, Japan.
English translation rights were arranged with
Shokabo Publishing Co., Ltd., through the
Japan Foreign-Rights Centre.
© 1990, 1987 Takashi Ono
Plenum Press is a Division of
Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
Preface
This book is a translation of my book Suron Josetsu (An
Introduction to Number Theory), Second Edition, published by
Shokabo, Tokyo, in 1988. The translation is faithful to the original
globally but, taking advantage of my being the translator of my
own book, I felt completely free to reform or deform the original
locally everywhere.
When I sent T. Tamagawa a copy of the First Edition of the
original work two years ago, he immediately pointed out that I had
skipped the discussion of the class numbers of real quadratic fields
in terms of continued fractions and (in a letter dated 2/15/87)
sketched his idea of treating continued fractions without writing
explicitly continued fractions, an approach he had first presented
in his number theory lectures at Yale some years ago. Although I
did not follow his approach exactly, I added to this translation a
section (Section 4.9), which nevertheless fills the gap pointed out
by Tamagawa. With this addition, the present book covers at least
T. Takagi's Shoto Seisuron Kogi (Lectures on Elementary Number
Theory), First Edition (Kyoritsu, 1931), which, in turn, covered at
least Dirichlet's Vorlesungen.
It is customary to assume basic concepts of algebra (up to,
say, Galois theory) in writing a textbook of algebraic number
theory. But I feel a little strange if I assume Galois theory and
prove Gauss quadratic reciprocity. Since it does not occupy much
space and time to introduce simple basic concepts of algebra and
since number theory is a rich source of structures of algebra, such as
groups, rings, and fields, I have included all definitions and proofs of
v
vi Preface
propositions in algebra except those of basic facts in linear algebra
such as linear independence, linear transformations, and deter
minants. In this way, the book covers the same basic facts of
algebraic number theory contained in Heeke's Vorlesungen (except
the quadratic reciprocity of an arbitrary number field) and in Part I
of Takagi's Daisuteki Seisuron (Algebraic Number Theory), First
Edition (Iwanami, 1948) (except the complete proof of Dedekind's
discriminant theorem).
Part II of Takagi's book is an exposition of the class field
theory based on the line of thought articulated in his fundamental
paper (1920) on abelian field extensions, simplified in several
places by the work of Artin, Hasse, and Chevalley in the early
1930s. A piece of good news for beginners is that the most
important map in the whole theory (the Artin map) is now within
their hands (Section 2.15 of this book). Guided by this map, they
can reach sufficient height to appreciate the beautiful flowers
discovered by those pioneers.
Takashi Ono
Baltimore
Contents
Notation and Conventions . . . . . . . . . . . . . .. xi
1. To the Gauss Reciprocity Law . 1
1.1. Basic Facts . . . . . . . 2
1.2. Modules in 7L . . . . . . 4
1.3. Euclidean Algorithm and Continued Fractions 8
1.4. Continued-Fraction Expansion of Irrational
Numbers. . . . . . . . . . . 12
1.5. Concept of Groups . . . . . . 16
1.6. Subgroups and Quotient Groups 21
1.7. Ideals and Quotient Rings . . . 23
1.8. Isomorphisms and Homomorphisms 25
1.9. Polynomial Rings . 28
1.10. Primitive Roots . . . . . . . 30
1.11. Algebraic Integers. . . . . . 34
1.12. Characters of Abelian Groups 37
1.13. The Gauss Reciprocity Law 41
2. Basic Concepts of Algebraic Number Fields . 44
2.1. Field Extensions . . . . . . 44
2.2. Galois Theory . . . . . . . 48
2.3. Norm, Trace, and Discriminant 53
2.4. Gauss Sum and Jacobi Sum. 55
vii
viii Contents
2.5. Construction of a Regular I-gon. . . 58
2.6. Subfields of the Ith Cyclotomic Field . 60
2.7. Cohomology of Cyclic Groups 63
2.8. Finite Fields . . . . . . . . . . . 68
2.9. Ring of Integers, Ideals, and Discriminant 69
2.10. Fundamental Theorem of Ideal Theory 74
2.11. Residue Class Rings . . . . . . . . . . 78
2.12. Decomposition of Primes in Number Fields . 81
2.13. Discriminant and Ramification 86
2.14. Hilbert Theory . . . . . . . . . . . . . 89
2.15. Artin Map . . . . . . . . . . . . . . . 93
2.16. Artin Maps of Subfields of the Ith Cyclotomic Field 97
2.17. The Artin Map in Quadratic Fields . . . . . .. 100
3. Analytic Methods. . . . . 105
3.1 Lattices in IRn. . . . . 105
3.2. Minkowski's Theorem 109
3.3. Dirichlet's Unit Theorem 113
3.4. Pre-Zeta Functions. . . 119
3.5. Dedekind Zeta Function 123
3.6. The mth Cyclotomic Field. 132
3.7. Djrichlet L-Functions. . . 134
3.8. Dirichlet's Theorem on Arithmetical Progressions 140
4. The lth Cyclotomic Field and Quadratic Fields 143
4.1. Determination of Gauss Sums . . . . . . 144
4.2. L-Functions and Gauss Sums . . . . . . 149
4.3. Class Numbers of Subfields of the Ith Cyclotomic
Field . . . . . . . . . . . . . . . 152
4.4. Class Number of Q(vT*) . . . . . . 155
4.5. Ideal Class Groups of Quadratic Fields 159
4.6. Cohomology of Quadratic Fields 166
4.7. Gauss Genus Theory . . . . . . . . 173
4.8. Quadratic Irrationals . . . . . . . . 179
4.9. Real Quadratic Fields and Continued Fractions 186
Contents ix
Answers and Hints to Exercises 195
Notes . . . . . . . . . . . . . . . . 210
A. Peano Axioms . . . . . . . . . 210
B. Fundamental Theorem of Algebra. 211
C. Zorn's Lemma . . . . . . . . . 212
D. Quadratic Fields and Quadratic Forms 212
List of Mathematicians. . . . . . . . . . . . . . . .. 215
Bih60graphy . . . . . . . . . . 216
Comments on the Bibliography . 220
Index . . . . . . . . . . . . . . . . . . . . . . .. 221
Notation and Conventions
Throughout this book, we use the standard symbols N, "l., 0, IR,
and C to represent natural numbers (i.e., positive integers),
integers, rational numbers, real numbers, and complex numbers,
respectively.
We assume that the reader is familiar with sets and the
symbols n, U, :::J, c, and E. If X, Yare sets, we write Xc Y to
mean that X is contained in Y but may possibly be equal to Y.
Similarly for X :::J Y. If f: X ~ Y is a mapping (or a map) from X
to Y, we write x ~ f(x) to denote the effect of f on x EX. We say
that f is injective if x =1= y implies f(x) =1= f(y). We say that f is
surjective if for any y E Y there exists x E X such that f(x) = y. We
say that f is bijective if it is both injective and surjective. We
denote by f(X) the set of all elements f(x), with x E X. In
symbols, we have f(X) = {f(x); X E X}. For a subset Y' c Y, we
put f-1(y,) = {x E X;f(x) E Y'}. We call f(X) the image of f and
f-1(y,) the inverse image of Y'. If g: Y ~ Z is another map, then
we have a composite map go f: X ~ Z such that (g 0 f)(x) =
g(f(x» for all x EX.
We use occasionally the symbols V, 3, and 3 to mean "for
1
any" (or "for all"), "there exists," and "there exists uniquely,"
respectively. If A and B are propositions, A='> B represents "A
implies B" and A ~ B represents "A if and only if B." When X is
a finite set, we denote by [Xl the cardinality of X. If the equality
sign "=" is used to define an object, we often write "<)g.,, We
assume that the reader is familiar with the equivalence relation in
a set X and the quotient set X / ~ defined by the relation ~.
xi
1
To the Gauss Reciprocity Law
This chapter consists of elementary number theory and deals with
the greatest common divisor, the euclidean algorithm, con
gruences, linear equations, primitive roots, and the quadratic
reciprocity law. The material covered here corresponds to the first
four chapters of Gauss's Disquisitiones arithmeticae (1801) and to
the whole volume (70 pages) of Weil's Number Theory for
Beginners (1985). Equally small (95 pages) Bakers A Concise
Introduction to the Theory of Numbers (1984) contains, in addition
to those standards, quadratic forms, diophantine approximation,
Fermat primes, Mersenne primes, Goldbach's conjecture, twin
primes, perfect numbers, the Riemann hypothesis, Euler's con
stant, C(2n + 1), Fermat's conjecture, Catalan's conjecture, and so
on. Following Weil, we regard this part of number theory as a rich
source of structures of algebra such as groups, rings, and fields.
Like computers in these days, the algebraic language is useful and
economizes our thoughts. For example, group theory teaches us
that the famous Fermat-Euler theorem on congruences is a special
case of a simple theorem on finite groups and that the important
existence theorem of primitive roots due to Gauss follows from the
theorem stating that a finite subgroup of the multiplicative group
of any field is cyclic. By doing so, number theory (elementary or
advanced) is well ventilated. As for the proof of the reciprocity
law, we followed the method of Gauss using Gauss sums because it
is not only beautiful but also has a crucial influence on the later
development of algebraic number theory.
1