Table Of ContentUndergraduate Texts in Mathematics
Editors
S. Axler
F. W. Gehring
K.A. Ribet
Springer Science+Business Media, LLC
Undergraduate Texts in Mathematics
Abbott: Understanding Analysis. Childs: A Concrete Introduction to
Anglin: Mathematics: A Concise History Higher Algebra. Second edition.
and Philosophy. Chung: Elementary Probability Theory
Readings in Mathematics. with Stochastic Processes. Third
Anglin/Larnbek: The Heritage of edition.
Thales. Cox/Little/O'Shea: Ideals, Varieties,
Readings in Mathematics. and Aigorithms. Second edition.
Apostol: Introduction to Analytic Croorn: Basic Concepts of Aigebraic
Number Theory. Second edition. Topology.
Arrnstrong: Basic Topology. Curtis: Linear Algebra: An Introductory
Arrnstrong: Groups and Symmetry. Approach. Fourth edition.
Axler: Linear Algebra Done Right. DevIin: The Joy of Sets: Fundamentals
Second edition. of Contemporary Set Theory.
Beardon: Limits: A New Approach to Second edition.
Real Analysis. Dixmier: General Topology.
BaklNewrnan: Complex Analysis. Driver: Why Math?
Second edition. Ebbinghaus/FlumlThomas:
Banchoff/Werrner: Linear Algebra Mathematical Logic. Second edition.
Through Geometry. Second edition. Edgar: Measure, Topology, and Fractal
Berberian: A First Course in Real Geometry.
Analysis. EIaydi: An Introduction to Difference
Bix: Conics and Cubics: A Equations. Second edition.
Concrete Introduction to Algebraic Erdös/Suninyi: Topics in the Theory of
Curves. Numbers.
Brernaud: An Introduction to Estep: Practical Analysis in One Variable.
Probabilistic Modeling. Exner: An Accompaniment to Higher
Bressoud: Factorization and Primality Mathernatics.
Testing. Exner: Inside Calculus.
Bressoud: Second Year Calculus. Fine/Rosenberger: The Fundamental
Readings in Mathematics. Theory of Algebra.
Brickrnan: Mathematicallntroduction Fischer: Intermediate Real Analysis.
to Linear Programming and Game Flanigan/Kazdan: Calculus Two: Linear
Theory. and Nonlinear Functions. Second
Browder: Mathematical Analysis: edition.
An Introduction. Fleming: Functions of Several Variables.
Buchmann: Introduction to Second edition.
Cryptography. Foulds: Combinatorial Optimization for
Buskes/van Rooij: Topological Spaces: Undergraduates.
From Distance to Neighborhood. Foulds: Optirnization Techniques: An
Callahan: The Geometry of Spacetime: Introduction.
An Introduction to Special and General FrankIin: Methods ofMathematical
Relavitity. Economics.
Carter/van Brunt: The Lebesgue Frazier: An Introduction to Wavelets
Stieltjes Integral: A Practical Through Linear Algebra.
Introduction.
Cederberg: A Course in Modem
Geometries. Second edition.
(continued after index)
Jobn StillweIl
Eleßlents of N ußlber
Theory
With 35 figures
Springer
John Sti1lwell
Mathematics Department
University of San Francisco
2130 FultoR Street
San Francisco. CA 94117~1080
USA
[email protected]
EJilflriu[ B(jard
s.
Axler F.W. Gchring K.A, Ribet
Mathematics DL"Partment Malhcmatics Dcpartmcnt Mathematics Departmcnt
San Francisco State Eallt Hali University ofCalifomia.
Univcrsity University of Michigan Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720·3840
USA USA USA
[email protected] fgehrin~mllth.llliI.umich.edu [email protected]:y.edu
ISBN 978-1-4419-3066-8 ISBN 978-0-387-21735-2 (eBook)
DOI 10.1007/978-0-387-21735-2
Mathcmatics Subjcct ClassifiClllion (2000): il-III
or
library Congrcss Cataloging-in-!>ublication Data
Stil1wcl1, John.
E1crnents ofnumber tbeory f John Slillwel1.
p. cm. - (1lndeqvaduate tcxlţ in mathematics)
Includcs bibliuwaphical references ane! index.
QA241 .SRI) 2002
512'.7- Je21 2002030569
Prinled o:: acid-frec paper.
@ 2003 Springer Scîence+Business Media New York
Originally publîshed by Springer-Verlag New York. lnc in 2003
Softcover reprint of tbe hardcover Ist edition 2003
AII rights reserved. This work lllay not be trans1ated or copied in wbole or in part WItbout tbe written
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ToElaine
Preface
This book is intended to complement my Elements oi Algebra, and it
is similarly motivated by the problem of solving polynomial equations.
However, it is independent of the algebra book, and probably easier. In
Elements oi Algebra we sought solution by radicals, and this led to the
concepts of fields and groups and their fusion in the celebrated theory of
Galois. In the present book we seek integer solutions, and this leads to the
concepts of rings and ideals which merge in the equally celebrated theory
of ideals due to Kummer and Dedekind.
Solving equations in integers is the central problem of number theory,
so this book is truly a number theory book, with most of the results found
in standard number theory courses. However, numbers are best understood
through their algebraic structure, and the necessary algebraic concepts
rings and ideals-have no better motivation than number theory.
The first nontrivial examples of rings appear in the number theory
of Euler and Gauss. The concept of ideal-today as routine in ring the
ory as the concept of normal subgroup is in group theory-also emerged
from number theory, and in quite heroic fashion. Faced with failure of
unique prime factorization in the arithmetic of certain generalized "inte
gers" , Kummer created in the 1840s a new kind of number to overcome
the difficulty. He called them "ideal numbers" because he did not know
exactly what they were, though he knew how they behaved. Dedekind in
1871 found that these "ideal numbers" could be realized as sets of actual
numbers, and he called these sets ideals.
Dedekind found that ideals could be defined quite simply; so much so
that a student meeting the concept today might wonder what all the fuss
is about. It is only in their role as "ideal numbers", where they realize
Kummer's impossible dream, that ideals can be appreciated as a genuinely
brilliant idea.
vii
viii Preface
Thus solution in integers-like solution by radicals-is a superb set
ting in which to show algebra at its best. It is the right place to introduce
rings and ideals and the right place first to apply them. It even gives an
opportunity to introduce some exotic rings, such as the quatemions, which
we use to prove Lagrange's theorem that every natural number is the sum
of four squares.
The book is based on two short courses (about 20 lectures each) given
at Monash University in recent years; one on elementary number theory
and one on ring theory with applications to algebraic number theory. Thus
the amount of material is suitable for a one-semester course, with some
variation possible through omission of the optional starred sections. A
slower-paced course could stop at the end of Chapter 9, at which point
most ofthe standard results have been covered, from Euclid's theorem that
there are infinitely many primes to quadratic reciprocity.
It should be stressed, however, that this is not meant to be a standard
number theory course. I have tried to avoid the ad hoc proofs that once
gave number theory a bad name, in favor of unifying ideas that work in
many situations. These inc1ude algebraic structures but also ideas from
elementary number theory, such as the Euc1idean algorithm and unique
prime factorization. In particular, I use the Euclidean algorithm as a bridge
to Conway's visual theory of quadratic forms, which offers a new approach
to the Pell equation.
There are exercises at the end of almost every section, so that each
new idea or proof receives immediate reinforcement. Some of them focus
on specific ideas, while others recapitulate the generalline of argument (in
easy steps) to prove a similar result. The purpose of each exercise should be
c1ear from the accompanying commentary, so instructors and independent
readers alike will be able to find an enjoyable path through the book.
My thanks go to the Monash students who took the courses on which
the book is based. Their reactions have helped improve the presentation in
many ways. I am particularly grateful to Ley Wilson, who showed that it
is possible to master the book by independent study.
Special thanks go to my wife Elaine, who proofread the first version
of the book, and to John Miller and Abe Shenitzer, who carefully read the
revised version and saved me from many mathematical and stylistic errors.
JOHN STILLWELL
South Melbourne, July 2002
Contents
Preface vii
1 Natural numbers and integers 1
1.1 Natural numbers . 2
1.2 Induction . . . . . . . . 3
1.3 Integers . . . . . . . . . 5
1.4 Division with remainder 7
1.5 Binary notation . . . . . 8
1.6 Diophantine equations 11
1.7 The Diophantus chord method 14
1.8 Gaussian integers 17
1.9 Discussion...... 20
2 The Euclidean algorithm 22
2.1 The gcd by subtraction . . . . . . . 22
2.2 The gcd by division with remainder 24
2.3 Linear representation of the gcd .. 26
2.4 Primes and factorization ..... . 28
2.5 Consequences of unique prime factorization 30
2.6 Linear Diophantine equations. . . 33
2.7 *The vector Euclidean algorithm 35
2.8 *The map of relatively prime pairs 38
2.9 Discussion. . . . . 40
3 Congruence arithmetic 43
3.1 Congruence mod n .......... . 44
3.2 Congruence c1asses and their arithmetic 45
3.3 Inverses mod p . . . . . ..... 48
ix
x Contents
3.4 Fennat's litde theorem ............ . 51
3.5 Congruence theorems of Wilson and Lagrange . 53
3.6 Inverses mod k ...... .. . 55
3.7 Quadratic Diophantine equations 57
3.8 *Primitive roots ...... . 59
3.9 *Existence of primitive roots 62
3.10 Discussion ..... 63
4 The RSA cryptosystem 66
4.1 Trapdoor functions 66
4.2 Ingredients of RSA 69
4.3 Exponentiation mod n . 70
4.4 RSA encryption and decryption . 72
4.5 Digital signatures ..... 73
4.6 Other computational issues 74
4.7 Discussion. . 74
5 The Pell equation 76
5.1 Side and diagonal numbers 77
2l
5.2 The equation:xl - = 1 78
5.3 The group of solutions .. 80
5.4 The general Pell equation and Z[vIn] 81
5.5 The pigeonhole argument ... 84
5.6 *Quadratic fonns . . . . . . . . . 87
5.7 *The map of primitive vectors .. 90
nl
5.8 *Periodicity in the map of:xl - 95
5.9 Discussion. . . . 99
6 The Gaussian integers 101
6.1 Z[i] and its norm ......... . 102
6.2 Divisibility and primes in Z[i] and Z 103
6.3 Conjugates ........ . 105
6.4 Division in Z[i] . . . . . . . 107
6.5 Fennat's two square theorem 109
6.6 Pythagorean tripies . . . . 110
+
6.7 *Primes of the fonn 4n 1 113
6.8 Discussion......... 115
