Problems in Algebraic Number Theory

353 Pages·2004·6.076 MB·English

by  Jody Esmonde

190
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
M. Ram Murty
Jody Esmonde
Problems in
Algebraic Number Theory
Second Edition
M.RamMurty JodyEsmonde
DepartmentofMathematicsandStatistics GraduateSchoolofEducation
Queen’sUniversity UniversityofCaliforniaatBerkeley
Kingston,OntarioK7L3N6 Berkeley,CA94720
Canada USA
[e-mail] [e-mail]
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment MathematicsDepartment
SanFranciscoState EastHall UniversityofCalifornia,
University UniversityofMichigan Berkeley
SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840
USA USA USA
[e-mail] [e-mail]. [e-mail]
umich.edu
MathematicsSubjectClassification(2000):11Rxx11-01
LibraryofCongressCataloging-in-PublicationData
Esmonde,Jody
Problemsinalgebraicnumbertheory/JodyEsmonde,M.RamMurty.—2nded.
p. cm.—(Graduatetextsinmathematics;190)
Includesbibliographicalreferencesandindex.
ISBN0-387-22182-4(alk.paper)
1.Algebraicnumbertheory—Problems,exercises,etc. I.Murty,MarutiRam. II.Title.
III.Series.
QA247.E76 2004
512.7′4—dc22 2004052213
ISBN0-387-22182-4 Printedonacid-freepaper.
©2005SpringerScience+BusinessMedia,Inc.
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It is practice first and knowledge afterwards.
Vivekananda
Preface to the Second
Edition
Since arts are more easily learnt by examples than precepts, I have
thought fit to adjoin the solutions of the following problems.
Isaac Newton, in Universal Arithmetick
Learningisamysteriousprocess. Noonecansaywhatthepreciserules
of learning are. However, it is an agreed upon fact that the study of good
examplesplaysafundamentalroleinlearning. Withrespecttomathemat-
ics,itiswell-knownthatproblem-solvinghelpsoneacquireroutineskillsin
howandwhentoapplyatheorem. Italsoallowsonetodiscovernuancesof
thetheoryandleadsonetoaskfurtherquestionsthatsuggestnewavenues
of research. This principle resonates with the famous aphorism of Lichten-
berg, “What you have been obliged to discover by yourself leaves a path in
your mind which you can use again when the need arises.”
This book grew out of various courses given at Queen’s University be-
tween1996and2004. Intheshortspanofasemester,itisdifficulttocover
enough material to give students the confidence that they have mastered
some portion of the subject. Consequently, I have found that a problem-
solving format is the best way to deal with this challenge. The salient
featuresofthetheoryarepresentedinclassalongwithafewexamples,and
then the students are expected to teach themselves the finer aspects of the
theory through worked examples.
Thisisarevisedandexpandedversionof“ProblemsinAlgebraicNum-
ber Theory” originally published by Springer-Verlag as GTM 190. The
new edition has an extra chapter on density theorems. It introduces the
readertothemagnificentinterplaybetweenalgebraicmethodsandanalytic
methods that has come to be a dominant theme of number theory.
IwouldliketothankAlinaCojocaru,WentangKuo,Yu-RuLiu,Stephen
Miller, Kumar Murty, Yiannis Petridis and Mike Roth for their corrections
and comments on the first edition as well as their feedback on the new
material.
Kingston, Ontario Ram Murty
March 2004
vii
Preface to the First
Edition
It is said that Ramanujan taught himself mathematics by systematically
working through 6000 problems1 of Carr’s Synopsis of Elementary Results
in Pure and Applied Mathematics. Freeman Dyson in his Disturbing the
Universe describes the mathematical days of his youth when he spent his
summermonthsworkingthroughhundredsofproblemsindifferentialequa-
tions. Ifwelookbackatourownmathematicaldevelopment,wecancertify
thatproblemsolvingplaysanimportantroleinthetrainingoftheresearch
mind. In fact, it would not be an exaggeration to say that the ability to
do research is essentially the art of asking the “right” questions. I suppose
Po´lya summarized this in his famous dictum: if you can’t solve a problem,
then there is an easier problem you can’t solve – find it!
This book is a collection of about 500 problems in algebraic number
theory. Theyaresystematicallyarrangedtorevealtheevolutionofconcepts
and ideas of the subject. All of the problems are completely solved and
no doubt, the solutions may not all be the “optimal” ones. However, we
feel that the exposition facilitates independent study. Indeed, any student
with the usual background of undergraduate algebra should be able to
work through these problems on his/her own. It is our conviction that
the knowledge gained by such a journey is more valuable than an abstract
“Bourbaki-style” treatment of the subject.
How does one do research? This is a question that is on the mind of
every graduate student. It is best answered by quoting Po´lya and Szego¨:
“General rules which could prescribe in detail the most useful discipline
of thought are not known to us. Even if such rules could be formulated,
they would not be very useful. Rather than knowing the correct rules of
thought theoretically, one must have them assimilated into one’s flesh and
blood ready for instant and instinctive use. Therefore, for the schooling of
one’s powers of thought only the practice of thinking is really useful. The
1Actually,Carr’sSynopsis isnotaproblembook. Itisacollectionoftheoremsused
bystudentstopreparethemselvesfortheCambridgeTripos. Ramanujanmadeitfamous
byusingitasaproblembook.
ix
x Preface
independent solving of challenging problems will aid the reader far more
than aphorisms.”
Asking how one does mathematical research is like asking how a com-
poser creates a masterpiece. No one really knows. However, it is clear
that some preparation, some form of training, is essential for it. Jacques
Hadamard, in his book The Mathematician’s Mind, proposes four stages
in the process of creation: preparation, incubation, illumination, and ver-
ification. The preparation is the conscious attention and hard work on a
problem. This conscious attention sets in motion an unconscious mecha-
nism that searches for a solution. Henri Poincar´e compared ideas to atoms
thataresetinmotionbycontinuedthought. Thedanceoftheseideasinthe
crucible of the mind leads to certain “stable combinations” that give rise
to flashes of illumination, which is the third stage. Finally, one must verify
the flash of insight, formulate it precisely, and subject it to the standards
of mathematical rigor.
This book arose when a student approached me for a reading course on
algebraicnumbertheory. Ihadbeenthinkingofwritingaproblembookon
algebraicnumbertheoryandItooktheoccasiontocarryoutanexperiment.
Itoldthestudenttoroundupmorestudentswhomaybeinterestedandso
sherecruitedeightmore. Eachstudentwouldberesponsibleforonechapter
of the book. I lectured for about an hour a week stating and sketching the
solution of each problem. The student was then to fill in the details, add
tenmoreproblemsandsolutions,andthentypesetitintoTEX. Chapters1
to 8 arose in this fashion. Chapters 9 and 10 as well as the supplementary
problems were added afterward by the instructor.
Some of these problems are easy and straightforward. Some of them
are difficult. However, they have been arranged with a didactic purpose.
It is hoped that the book is suitable for independent study. From this
perspective,thebookcanbedescribedasafirstcourseinalgebraicnumber
theory and can be completed in one semester.
Ourapproachinthisbookisbasedontheprinciplethatquestionsfocus
the mind. Indeed, quest and question are cognates. In our quest for truth,
for understanding, we seem to have only one method. That is the Socratic
method of asking questions and then refining them. Grappling with such
problems and questions, the mind is strengthened. It is this exercise of the
mind that is the goal of this book, its raison d’ˆetre. If even one individual
benefits from our endeavor, we will feel amply rewarded.
Kingston, Ontario Ram Murty
August 1998
Acknowledgments
We would like to thank the students who helped us in the writing of this
book: Kayo Shichino, Ian Stewart, Bridget Gilbride, Albert Chau, Sindi
Sabourin, Tai Huy Ha, Adam Van Tuyl and Satya Mohit.
WewouldalsoliketothankNSERCforfinancialsupportofthisproject
as well as Queen’s University for providing a congenial atmosphere for this
task.
J.E.
M.R.M.
August 1998
xi
Contents
Preface to the Second Edition vii
Preface to the First Edition ix
Acknowledgments xi
I Problems
1 Elementary Number Theory 3
1.1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Applications of Unique Factorization . . . . . . . . . . . . . 8
1.3 The ABC Conjecture . . . . . . . . . . . . . . . . . . . . . 9
1.4 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 10
2 Euclidean Rings 13
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Gaussian Integers . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Eisenstein Integers . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Some Further Examples . . . . . . . . . . . . . . . . . . . . 21
2.5 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 25
3 Algebraic Numbers and Integers 27
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Liouville’s Theorem and Generalizations . . . . . . . . . . . 29
3.3 Algebraic Number Fields. . . . . . . . . . . . . . . . . . . . 32
3.4 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 38
4 Integral Bases 41
4.1 The Norm and the Trace. . . . . . . . . . . . . . . . . . . . 41
4.2 Existence of an Integral Basis . . . . . . . . . . . . . . . . . 43
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Ideals in O . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
K
4.5 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 50
xiii
xiv CONTENTS
5 Dedekind Domains 53
5.1 Integral Closure . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Characterizing Dedekind Domains . . . . . . . . . . . . . . 55
5.3 Fractional Ideals and Unique Factorization. . . . . . . . . . 57
5.4 Dedekind’s Theorem . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Factorization in O . . . . . . . . . . . . . . . . . . . . . . 65
K
5.6 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 66
6 The Ideal Class Group 69
6.1 Elementary Results . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Finiteness of the Ideal Class Group . . . . . . . . . . . . . . 71
6.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . 73
6.4 Exponents of Ideal Class Groups . . . . . . . . . . . . . . . 75
6.5 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 76
7 Quadratic Reciprocity 81
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 The Law of Quadratic Reciprocity . . . . . . . . . . . . . . 86
7.4 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . 88
7.5 Primes in Special Progressions . . . . . . . . . . . . . . . . 91
7.6 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 94
8 The Structure of Units 99
8.1 Dirichlet’s Unit Theorem . . . . . . . . . . . . . . . . . . . 99
8.2 Units in Real Quadratic Fields . . . . . . . . . . . . . . . . 108
8.3 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 115
9 Higher Reciprocity Laws 117
9.1 Cubic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . 117
9.2 Eisenstein Reciprocity . . . . . . . . . . . . . . . . . . . . . 122
9.3 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 125
10 Analytic Methods 127
10.1 The Riemann and Dedekind Zeta Functions . . . . . . . . . 127
10.2 Zeta Functions of Quadratic Fields . . . . . . . . . . . . . . 130
10.3 Dirichlet’s L-Functions . . . . . . . . . . . . . . . . . . . . . 133
10.4 Primes in Arithmetic Progressions . . . . . . . . . . . . . . 134
10.5 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 136
11 Density Theorems 139
11.1 Counting Ideals in a Fixed Ideal Class . . . . . . . . . . . . 139
11.2 Distribution of Prime Ideals . . . . . . . . . . . . . . . . . . 146
11.3 The Chebotarev density theorem . . . . . . . . . . . . . . . 150
11.4 Supplementary Problems. . . . . . . . . . . . . . . . . . . . 153