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Factorization
Unique and Otherwise
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CMS Treatises in Mathematics
Publishedby the CanadianMathematicalSociety
Traite´s de mathe´matiques de la SMC
Publie´ par la Socie´te´ mathe´matiquedu Canada
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Jonathan Borwein
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Factorization
Unique and Otherwise
Steven H. Weintraub
Lehigh University
CanadianMathematicalSociety AKPeters,Ltd.
Socie´te´ mathe´matiqueduCanada Wellesley,Massachusetts
Ottawa,Ontario
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Library of Congress Cataloging-in-Publication Data
Weintraub,StevenH.
Factorization : uniqueand otherwise / Morgens Esrom Larsen.
p. cm. -- (CMS Treatises in mathematics)
Includesindex.
ISBN 978-1-56881-241-0 (alk. paper)
1. Factorization (Mathematics). 2. Rings of integers. 3. Rings
(Algebra). I. Title.
QA161.F3W45 2008
512.7(cid:2)2--dc22
2007049328
Printed in Canada
12 11 10 09 08 10 9 8 7 6 5 4 3 2 1
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To my nephews, nieces, and grandkids:
Wendy, Jenny, and William;
Erica, Jordan, and Allison;
Blake, Natalie, and Ethan
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Contents
Preface ix
Introduction 1
1 BasicNotions 7
1.1 Integral Domains . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 UniqueFactorization 19
2.1 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . 20
2.2 The GCD-L Property and Euclid’s Algorithm . . . . . . . 31
2.3 Ideals and Principal Ideal Domains . . . . . . . . . . . . . 45
2.4 Unique Factorization Domains . . . . . . . . . . . . . . . 51
2.5 Nonunique Factorization: The Case D <0 . . . . . . . . . 60
2.6 Nonunique Factorization: The Case D >0 . . . . . . . . . 67
2.7 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 TheGaussianIntegers 91
3.1 Fermat’s Theorem . . . . . . . . . . . . . . . . . . . . . . 92
3.2 Factorization into Primes . . . . . . . . . . . . . . . . . . 101
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 Pell’sEquation 111
4.1 Representations and Their Composition . . . . . . . . . . 112
4.2 Solving Pell’s Equation. . . . . . . . . . . . . . . . . . . . 118
4.3 Numerical E√xamples and Further Results . . . . . . . . . 127
4.4 Units in O( D) . . . . . . . . . . . . . . . . . . . . . . . 137
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
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viii Contents
5 TowardsAlgebraicNumberTheory 143
5.1 Algebraic Numbers and Algebraic Integers . . . . . . . . . 144
5.2 Ideal Theory . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . 150
5.4 Algebraic Number√Fields and Dedekind Domains . . . . . 154
5.5 Prime Ideals in O( D) √. . . . . . . . . . . . . . . . . . . 158
5.6 Examples of Ideals in O( D) . . . . . . . . . . . . . . . . 166
5.7 Behavior of Ideals in Algebraic Number Fields . . . . . . 178
5.8 Ideal Elements . . . . . . . . . . . . . . . . . . . . . . . . 180
5.9 Dirichlet’s Unit Theorem . . . . . . . . . . . . . . . . . . 182
5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A MathematicalInduction 191
A.1 Mathematical Induction and Its Equivalents . . . . . . . . 191
A.2 Consequences of Mathematical Induction . . . . . . . . . 196
A.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
B Congruences 205
B.1 The Notion of Congruence . . . . . . . . . . . . . . . . . . 205
B.2 Linear Congruences. . . . . . . . . . . . . . . . . . . . . . 211
B.3 Quadratic Congruences . . . . . . . . . . . . . . . . . . . 223
B.4 Proof of the Law of Quadratic Reciprocity . . . . . . . . . 236
B.5 Primitive Roots . . . . . . . . . . . . . . . . . . . . . . . . 241
B.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
C ContinuationsfromChapter2 251
C.1 Continuation of the Proof of Theorem 2.8 . . . . . . . . . 251
C.2 Continuation of Example 2.26 . . . . . . . . . . . . . . . . 255
C.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Index 259
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Preface
In this book, we introduce the reader to some beautiful and interesting
mathematics, which is not only historically important but also still very
much alive today. Indeed, it plays a central role in modern mathematics.
The mathematical content of this book is outlined in the introduction,
but we shall preview it here. It is a basic property of the integers, known
as the Fundamental Theoremof Arithmetic, that every integer can be fac-
tored into a product of primes in an essentially unique way. Our principal
objective in this book is to investigate somewhat more general but still
relatively concrete systems (known as rings of integers in quadratic fields)
andseewhenthis propertydoesordoesnotholdforthem. Weaccomplish
this objective in Chapters 1 and 2. But this investigation naturally leads
us into further investigations—mathematics is like that—and we consider
related questions in Chapters 3 and 4, where we investigate the Gaussian
integers and Pell’s equation, respectively.
The questions we investigatehere were atthe roots ofthe development
of algebraic number theory. In Chapter 5 we provide an overview of alge-
braic number theory with emphasis on how the results for quadratic fields
generalize to arbitrary algebraic number fields.
We envision several ways in which this book can be used. One way
is for a first course in number theory. In our investigations, we begin at
the beginning, so this book is suitable for that purpose. Indeed, one of
the themes of this book is that one can go a long ways with only ele-
mentary methods. To be sure, the topics covered here are not the tradi-
tional topics for a first course in number theory (though there is consid-
erable overlap), but there is no reason that the traditional topics need be
sacrosanct.
Another way to use this book is for a more advanced course in number
theory, and there is plenty of appropriate material here for such a course.
Indeed, thereisfarmorethanasemester’sworthofmaterialhere,evenfor
an advanced course.
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x Preface
Inthisregard,wecallthereader’sattentiontoAppendicesAandB,on
mathematicalinduction andcongruences,respectively. Ifthis book is used
as a text for a first course, much of the material in these two appendices
shouldbecovered. Ifthisbookisusedasatextforamoreadvancedcourse,
these appendices will serve as background.
We have not tried to write a textbook on algebraic number theory in
Chapter 5, but rather to provide an overviewof the field. But we feel that
this overview can serve as a valuable introduction to, and guide for, the
student who wishes to study this field, and can also serve as a concrete
reference for some of the general results that a student of this field will
encounter.
Steven H. Weintraub
Bethlehem, PA, USA
August 2007
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Introduction
We shall here be concerned with the circle of ideas that surrounds the
Fundamental Theorem of Arithmetic.
First we recall the usual definition of a prime: a prime number is a
positive integer, other than 1, that has no divisors except itself and 1. For
example 2 and 3 are primes, but 6=2·3 and 10=2·5 are not.
Then the Fundamental Theorem of Arithmetic states that every posi-
tive integer can be factored into primes in an essentially unique way. For
example,
1=1,
2=2,
6=2·3,
10=2·5,
15=3·5,
2499=3·72·17.
By“essentiallyunique,” wemeanunique uptothe orderofthe factors,
so that we consider 6 = 2·3 = 3·2 to be the same factorization. (Note
that 1 is a specialcase. We think ofit as havingan“empty” factorization,
as it is not divisible by any prime.)
As its name implies, unique factorization is a fundamental property of
thepositiveintegers,apropertythatwasknowntotheancientGreeks. We
will prove this property, and indeed our proof will follow that of Euclid.
But we will be interested in examining this proof and seeing what makes
it really “work,” with an idea of seeing when we can extend it to more
general situations.
√
For example, let us considernumbers ofthe forma+b −1 with a and
b integers. It turns out, and we shallprove,that numbers of this form also
haveuniquefactorization. Forexample,wehavethefollowingfactorization
1
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