Number Theory (Pure and Applied Mathematics)

449 Pages·1986·16.95 MB·English

by  Z. I. Borevich

Number Theory
PURE AND APPLIED MATHE M AT IC S
A Series of Monographs and Textbooks
Edited by
PAULA . SMITHan d SAMUEELI LENBERG
Columbia University, New York
1 : ARNOLDSO MMERFEPLarDti.a l Differential Equations in Physics. 1949 (Lectures
on Theoretical Physics, Volume VI)
2 : REINHOLDB AER.L inear Algebra and Projective Geometry. 1952
3 : HERBERBT USEMANANN D PAULK ELLY.P rojective Geometry and Projective
Metrics. 1953
4: STEFABNE RCMAANN D M. SCHIFFEKRer. nel Functions and Elliptic Differential
Equations in Mathematical Physics. 1953
5 : RALPHP HILIBPO AS,J R.E ntire Functions. 1954
6: HERBEBRUTS EMANTNh. e Geometry of Geodesics. 1955
7 : CLAUDEC HEVALLEFYu. ndamental Concepts of Algebra. 1956
8: SZE-TSENH u. Homotopy Theory. 1959
9: A. M. OSTROWSKSI.o lution of Equations and Systems of Equations. Second
Edition. 1966
10 : J. DIEUDONNFo~u.n dations of Modern Analysis. 1960
11 : S. I. GOLDBERCGu. rvature and Homology. 1962
12 : SICURDHUERL GASOND.i fferential Geometry and Symmetric Spaces. 1962
13 : T. H. HILDEBRANInDtroTd.u ction to the Theory of Integration. 1963
14: SHREERAAMBH YANKALRo.c al Analytic Geometry. 1964
15 : RICHARLD. BISHOPA ND RICHARJD. CRITTENDEGNe. ometry of Manifolds. 1964
16: STEVENA . GAAL.P oint Set Topology. 1964
17 : BARRYM ITCHELLT. heory of Categories. 1965
18: ANTHONYP . MORSE.A Theory of Sets. 1965
19: GUSTAVEC HOQUETT. opology. 1966
20: Z. I. BOREVICAHN D I. R. SHAFAREVICNHu. mber Theory. 1966
21 : Josf LUISM ASSERAA ND JUAN JORCE SCHAFFELRin.e ar Differential Equations
and Function Spaces. 1966
In preparation:
MARTINE ICHLER.I ntroduction to the Thecry of Algebraic Numbers and
Functions.
RICHARDD . SCHAFEARn. Introduction to Nonassociative Algebras.
FRANCOTIRSE VESTo. pological Vector Spaces, Distributions, and Kernels.
Number Theory
by
Z. I. BOREVICH
and I. R. SHAFAREVICH
Translated by NEWCOMB GREENLEAF for
SCRIPTA TECHNICA, Translators
New York, New York
W
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Harcourt Brnce Jownovich, Publishers
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COPYRIGHT 0 1966, BY ACADEMIPCR ESS INC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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ACADEMIC PRESS, INC.
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Translator’s Preface
This book was written as a text for the learning of number theory, not as
a reference work, and we have attempted to preserve the informal, slow-
placed style of the original. The emphasis of the book is on number theory
as a living branch of modern mathematics, rather than as a collection of
miscellaneous results.
The book should prove accessible to any advanced undergraduate in
mathematics, or to any graduate student. The reader should be familiar with
the basic concepts of abstract algebra, and should have followed analysis
through a standard advanced calculus course. While some results from ele-
mentary number theory are occasionally used, a previous course in number
theory is certainly not necessary, though the reader without such a course
may have a few occasions for consulting a more elementary text.
Almost all of the notation and terminology is standard. The only difficulty
arose in the terminology for valuations. Since there does not seem to be any
universally adopted terminology in English, we have, after some hesitation,
followed that of the authors, which has the advantage of being clear and
simple in this context. Thus we reserve the term “valuation” for the case
when the value group is the integers. Mappings into the positive reals are
called “metrics.”
We would like to mention some additional references. The theory of
quadratic forms receives a systematic development in the book Zntroduction
to Quadratic Forms, by 0.T . O’Meara (New York, 1963). In particular,
O’Meare presents a proof of the Hasse-Minkowski theorem which does not
use the Dirichlet theorem on primes in arithmetic progressions. In Chapter 7
of Commutative Algebre (Paris, 1965), Bourbaki gives a complete exposi-
tion of the theory of divisors, from a somewhat more abstract standpoint
than that found in Chapter 3.
NEWCOMGBR EENLEAF
Rochester, New York
V
This Page Intentionally Left Blank
This book is written for the student in mathematics. Its goal is to give a
view of the theory of numbers, of the problems with which this theory deals,
and of the methods that are used.
We have avoided that style which gives a systematic development of the
apparatus and have used instead a freer style, in which the problems and the
methods of solution are closely interwoven. We start from concrete prob-
lems in number theory. General theories arise as tools for solving these
problems. As a rule, these theories are developed sufficiently far so that the
reader can see for himself their strength and beauty, and so that he learns
to apply them.
Most of the questions that are examined in this book are connected with
the theory of diophantine equations - that is, with the theory of the solu-
tions in integers of equations in several variables. However, we also consider
questions of other types; for example, we derive the theorem of Dirichlet on
prime numbers in arithmetic progressions and investigate the growth of the
number of solutions of congruences.
The methods that we use are primarily algebraic. More precisely, we work
with finite field extensions and with metrics on them. However, analytic
methods have a considerable place. Chapter 5 is devoted to them, and p-adic
analytic functions are used in Chapter 4. Geometric concepts play a consid-
erable role in several spots.
The book does not presuppose a great deal of knowledge on the part of
the reader. For reading most of it, two university courses would be completely
satisfactory. Some facts on analytic functions are used in the last two
chapters.
The necessary prerequisites of an algebraic nature are given in the “Alge-
braic Supplement” at the end of the book. There the reader will find defini-
tions, results, and some proofs that are used in the book but might not
appear in a university course in higher algebra.
This book grew out of a course taught by one of the authors at Moscow
University. We would like to thank A. G. Postnikov, who allowed us the
use of his notes from this course.
vii
viii PREFACE
We are also extremely grateful to Dmitri Constantine Faddeev, who made
many contributions to this book. He should receive credit for some of the
proofs that appear in this book, for example, the new p-adic proof of the
theorem of Kummer on the second factor in the number of divisor classes of
a cyclotomic field.
Moscow THEA UTHORS
Trunslutor's Preface V
Foren'ord vii
Chapter 1.
Congruences 1
1. Congruences with Prime Modulus 3
2. Trigonometric Sums 9
3. p-Adic Numbers 18
4. An Axiomatic Characterization of the Field of p-adic Numbers 32
5. Congruences and p-adic Integers 40
6. Quadratic Forms with p-adic Coefficients 47
7. Rational Quadratic Forms 61
Chapter 2.
Representation of Numbers by Decomposable Forms 75
I. Decomposable Forms 77
2. Full Modules and Their Rings of Coefficients 83
3. Geometric Methods 94
4. The Groups of Units 107
5. The Solution of the Problem of the Representation of Rational
Numbers by Full Decomposable Forms 116
6. Classes of Modules 123
7. Representation of Numbers by Binary Quadratic Forms 129
ix