Cohn - Advanced Number Theory (1962).pdf

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ADVANCED
NUMBERTHEORY
Harvey Cohn
Distinguished
Professor of Mathematics
City University
of New York
Dover Publications, Inc.
NewYork
dedicated
to TONY
and SUSAN
Copyright
0 1962 by Harvey
Cohn.
All rights reserved
under Pan American
national
Copyright
Conventions.
Published
in
pany, Ltd., 30
tario.
Published
in
Company,
Ltd.,
and
Inter-
Canada
by General
Publishing
Lesmill Road, Don Mills, Toronto,
the United
10 Orange
Kingdom
by Constable
Street, London WC2H
Com-
On-
and
7EG.
This Dover edition, first published
in 1980, is an un-
abridged
and corrected
republication
of the work first
published
in 1962 by John Wiley & Sons, Inc., under the
title A
Second
Course
in Number
Theory.
International
Standard
Book
Library
of Congres
Catalog
Number:
0-486-64023-X
Cord Number:
BO-65862
Manufactured
in the United States of America
Dover Publications,
Inc.
100 Varick Street
New York. N.Y. 10014
PREFACE
The prerequisites for this book are the “standard” first-semester course
in number theory (with incidental elementary algebra) and elementary
calculus. There is no lack of suitable texts for these prerequisites (for
example, An Introduction to the Theory of Numbers, by 1. Niven and H. S.
Zuckerman, John Wiley and Sons, 1960, cari be cited as a book that intro-
duces the necessary algebra as part of number theory).
Usually, very little
else cari be managed in that first semester beyond the transition from
improvised combinatorial amusements of antiquity to the coherently
organized background for quadratic reciprocity, which was achieved in
the eighteenth Century.
The present text constitutes slightly more than enough for a second-
semester course, carrying the student on to the twentieth Century by
motivating some heroic nineteenth-Century developments in algebra and
analysis. The relation of this textbook to the great treatises Will necessarily
be like that of a hisforical novel to chronicles. We hope that once the
student knows what to seek he Will find “chronicles” to be as exciting as a
“historical novel.”
The problems in the text play a significant role and are intended to
stimulate the spirit of experimentation ivhich has traditionally ruled
number theory and which has indeed become resurgent with the realization
of the modern computer. A student completing this course should acquire
an appreciation for the historical origins of linear algebra, for the zeta-
function tradition, for ideal class structure, and for genus theory. These
V
vi
PREFACE
ideas,although relatively old, still make their influence felt on the frontiers
of modern mathematics. Fermat’slast theorem and complex multiplication
are unfortunate omissions,but the motive was not to depressthe degree
of difficulty SOmuch as it was to make the most efficient usageof one
semester.
My acknowledgmentsare many and are difficult to list. 1 enjoyed the
benefitsof coursesunder Bennington P. Gill at City Collegeand Saunders
MacLane at Harvard. The book profited directly from suggestions my
by
students and from the incidental advice of many readers, particularly
Burton W. Jonesand Louis J. Mordell. 1 owe a specialdebt to Herbert S.
Zuckerman for a careful reading, to Gordon Pal1for major improvements,
and to the staff of John Wiley and Sons for their cooperation.
HARVEY
COHN
Tucson,
October
Arizona
1961
.
CONTENTS
Note:
Tbe sections marked with * or ** might be omitted in class use
if there is a lack of time. (Here the ** sections are considered more truly
optional.)
INTRODUCTORY
SURVEY
Diophantine Equations 1
Motivating Problem in Quadratic Forms 2
Use of Algebraic Numbers 5
Primes in Arithmetic Progression 6
PART 1. BACKGROUND
MATERIAL
9
9
1. Review of Elementary Number Theory and Croup Theory
Number Theoretic Concepts
1. Congruence 9
2. Unique factorization
10
3. The Chinese remainder theorem 11
4. Structure of reduced residue classes
12
5. Residue classes for prime powers 13
Group Theoretic Concepts
6. Abelian groups and subgroups 15
7. Decomposition
into cyclic groups 16
Quadratic Congruences
8. Quadratic residues 18
9. Jacobi symbol 20
vii
15
18

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