Table Of Content190
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
[email protected] [email protected]
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
[email protected] [email protected]. [email protected]
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.
All rights reserved. This work may not be translated or copied in whole or in part without the
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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
P´olya 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
