Certain number-theoretic episodes in algebra

630 Pages·2007·2.303 MB·English

by  R Sivaramakrishnan

Certain
Number-Theoretic
Episodes in
Algebra
R. Sivaramakrishnan
Boca Raton London New York
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Library of Congress Cataloging‑in‑Publication Data
Sivaramakrishnan, R., 1936‑
Certain number‑theoretic episodes in algebra / R. Sivaramakrishnan.
p. cm. ‑‑ (Pure and applied mathematics ; 286)
Includes bibliographical references and indexes.
ISBN 0‑8247‑5895‑1 (alk. paper)
1. Algebraic number theory. 2. Number theory. I. Title. II. Series.
QA247.S5725 2006
512.7’4‑‑dc22 2006048994
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MONOGRAPHS AND TEXTBOOKS IN
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PURE AND APPLIED MATHEMATICS Recent Titles
E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, Second
Edition, Revised and Expanded (2004)
A Program of Monographs, Textbooks, and Lecture Notes
M. M. Rao, Measure Theory and Integration, Second Edition, Revised and Expanded
(2004)
W. J. Wickless, A First Graduate Course in Abstract Algebra (2004)
EXECUTIVE EDITORS
R. P. Agarwal, M. Bohner, and W-T Li, Nonoscillation and Oscillation Theory for
Functional Differential Equations (2004)
Earl J. Taft Zuhair Nashed
J. Galambos and I. Simonelli, Products of Random Variables: Applications to
Rutgers University University of Central Florida
Problems of Physics and to Arithmetical Functions (2004)
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Walter Ferrer and Alvaro Rittatore, Actions and Invariants of Algebraic Groups (2005)
Christof Eck, Jiri Jarusek, and Miroslav Krbec, Unilateral Contact Problems: Variational
Methods and Existence Theorems (2005)
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A. B. Kharazishvili, Strange Functions in Real Analysis, Second Edition (2006)
M. S. Baouendi Freddy van Oystaeyen
Vincenzo Ancona and Bernard Gaveau, Differential Forms on Singular Varieties:
University of California, University of Antwerp, De Rham and Hodge Theory Simplified (2005)
San Diego Belgium Santiago Alves Tavares, Generation of Multivariate Hermite Interpolating Polynomials
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Jane Cronin Donald Passman
Sergio Macías, Topics on Continua (2005)
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Madison
Jack K. Hale Contact Problems with Adhesion or Damage (2006)
Georgia Institute of Technology Fred S. Roberts Marwan Moubachir and Jean-Paul Zolésio, Moving Shape Analysis and Control:
Rutgers University Applications to Fluid Structure Interactions (2006)
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Alfred Geroldinger and Franz Halter-Koch, Non-Unique Factorizations: Algebraic,
University of California, David L. Russell
Combinatorial and Analytic Theory (2006)
Berkeley Virginia Polytechnic Institute
Kevin J. Hastings, Introduction to the Mathematics of Operations Research
Marvin Marcus and State University withMathematica®, Second Edition (2006)
University of California, Walter Schempp Robert Carlson, A Concrete Introduction to Real Analysis (2006)
Santa Barbara Universität Siegen John Dauns and Yiqiang Zhou, Classes of Modules (2006)
N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados,
W. S. Massey Mark Teply
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DEPARTMENTOFSCIENCEANDTECHNOLOGY
UNDERITS
UTILIZATIONOFSCIENTIFICEXPERTISEOFRETIREDSCIENTISTS
SCHEME
(USERSscheme)
PROJECTNo: HR/UR/21/98
The author acknowledges with thanks the (cid:2)nancial support of the
Department of Science and Technology under USERSscheme for under-
taking the project for preparation of a monograph/textbook. But for the
timely(cid:2)nancialhelpoftheDSTandtheencouraging lettersreceivedfrom
Dr. Parveen Farooqui, Head, Human Resources Wing of DST,the task of
implementation wouldnothavefoundful(cid:2)lment.
R.Sivaramakrishnan
© 2007 by Taylor & Francis Group, LLC
Dedicatedtothememoryof
myparents
(late)R.R.RAMAKRISHNA AYYAR(1907(cid:150)1985)
&
(late)T.S.LAKSHMYAMMAL(1911(cid:150)1975)
and
thefollowingteachers
(late)P.I.IKKORAN,P.SANKARANNAIR
&
S.PARAMESWARAIYER
(inhighschoolclasses)
(late)K.GOPALAPANICKER,T.S.RAMANATHAIYER,
N.P.SUBRAMANIAIYER,P.ACHUTHANPILLAI,
K.X.JOHN,N.P.INASU&V.KARUNAKARAMENON
(incollegeclasses)
and
(late)C.S.VENKATARAMAN(thesisadviser),
(late)P.KESAVAMENON,K.NAGESWARARAO&N.V.BEERAN
eachoneofwhomin(cid:3)uencedtheauthorinlearningaboutrigourin
mathematicstherightway.
© 2007 by Taylor & Francis Group, LLC
ACKNOWLEDGEMENT
Theauthorwishestoexpresshisdeepsenseofgratitudetotheauthoritiesofthe
UniversityofCalicutforhavinggivenhimtheopportunitytoutilizethefacilities
attheCalicutUniversityCampusingeneralandattheMathematicsDepartment
inparticular.
Thanksaredueto
(1) Prof. V.Krishnakumarforhishelpandguidanceataverypersonalleveland
inhiscapacityasHead,DepartmentofMathematics
(2) Dr. P. T. Ramachandran for valuable discussions and Sri. Kuttappan C,
Librarian in the Department of Mathematics for having provided abundant
helpinthematteroflibraryreference
(3) theDeputyRegistrar, Pl.D Branchandhiscolleaguesformattersofof(cid:2)cial
correspondence
(4) theFinanceOf(cid:2)cerandhisstaffforprocessingbills andvouchersandsuch
othertransactions.
Many academicians and friends have offered help at various stages in the
progressoftheproject.
The author had the opportunity to visit Mangalore University, Mangalore
(Karnataka)during 1996(cid:150)97. It was during this period that the spade work for
theprojectwasdone,andtheencouragementreceivedfromProf. B.G.Shenoy
and Prof. Juliet Britto was of great help. The author thanks the Faculty of the
DepartmentofMathematics,MangaloreUniversityfortheinvitationtostayand
workforayear.
Thanksaredueto
(i) Prof. C.S.SeshadriFRS,Director,ChennaiMathematicalInstitute,Chen-
naiforarrangingavisittothelibraryoftheInstitute
(ii) Prof. R.SridharanandProf. K.R.NagarajanofChennaiMathematicalIn-
stituteforadviceabouttheinclusionofcertaintopics/papersinthesections
onthePellequationandringswithchainconditions
(iii) Prof. R. Balasubramanian, Director, Institute of Mathematical Sciences
(MATSCIENCE) for valuable comments in the choice of topics in Alge-
braicNumberTheory
(iv) Prof. M.ThampanNair andProf. C. Ponnuswamyforhelpinconnection
withavisittoI.I.T(Madras),Chennai
(v) Prof. T.ThrivikramanandProf. R.S.ChakravartiofCochinUniversityof
ScienceandTechnology,Kochifortheirscholarlysuggestionsandremarks
(vi) Prof. M.I.JinnahandProf. A.R.Rajanforfacilitatingtheauthor’svisitto
theMathematicsDepartment,UniversityofKerala,Thiruvananthapuram
(vii) Dr. Rajendran Valiaveetil for all the help renderedby way of discussions
and for having assisted the author by going through the (cid:2)rst draft of the
manuscriptandformakingsuggestionsaboutthesimpli(cid:2)cationofproofsof
sometheorems
© 2007 by Taylor & Francis Group, LLC
(viii) Prof.PenttiHaukkanenoftheUniversityofTampere,Finlandand
Prof. S. A.Katre ofthe UniversityofPoona, Puneforsendingreprintsof
theirarticlestotheauthor
(ix) Prof. DonRedmondofSouthernIllinoisUniversity,Carbondaleforhaving
suppliedthereferencesrelatingtotheGoldbachconjecture
(x) Prof.V.K.Balachandran,formerDirector,RamanujanInstitutefor
Advanced Study, Chennai for help received by way of discussions and
correspondence
(xi) Dr.N.Raju,Head,DepartmentofStatistics,UniversityofCalicutfortimely
helpandadviceinthematteroftypesettingandformatofthemanuscript.
© 2007 by Taylor & Francis Group, LLC
PREFACE
Thismonographisanattempttojustifythefollowingassertion: (cid:147)Itisdesir-
abletolearnalgebravianumbertheoryandtolearnnumbertheoryviaalgebra(cid:148).
Many concepts in commutative algebra such as Euclidean domains, prime
andprimaryideals,(cid:2)eldofquotientsofanintegraldomainandsuchothershave
originatedfromnotionsinnumbertheory.Foronewhogoesdeeperintothe(cid:2)ner
aspectsofnumbertheory,algebraictechniqueswouldappeartobepowerfuland
elegant.ExamplesarefromthecrispproofsofGauss’squadraticreciprocitylaw,
Fermat’sTwo-squarestheoremandLagrange’stheoremontheexpressibilityofa
positiveintegerasasumoffoursquares.Thoughallauthorsofbooksonnumber
theoryhaveemphasizedthisaspect,perhaps,twobooksthatmakethealgebraic
approachexplicitare
1. EthanD.Bolker:ElementaryNumberTheory(cid:151)anAlgebraicApproach
W.A.BenjaminInc.NY(1970)and
2. F.Richman:NumberTheory(cid:151)AnIntroductiontoAlgebra
Brooks/ColeMonterey/California(1971).
ItistruethatclassicaltextbookssuchasO.ZariskiandP.Samuel:
Commutative Algebra Vols I and II (SpringerVerlag GTM Nos. 28, 29 (1982)
original versionVan Nostrand Edition (1958))and K. Ireland and M. I. Rosen:
AClassicalIntroductiontoModernNumberTheory,2ndEdition,SpringerVerlag
GTM No. 84 (1985) original version: Bogden and Quigley Inc., Publishers,
Tarrytown-on-Hudson,NY(1972)conveythemessageofdoingalgebrawithfull
number-theoreticsupportandviceversaexceedinglywell.
Theaimofthismonographistospreadthismessagewithgreateremphasis.
It is forthe mathematicalcommunity,at large, to pass judgementas to howfar
thedesiredgoalhasbeenachieved.
Thismonographpresupposesrudimentaryknowledgeofelementarynumber
theoryaswellasalgebraonthepartofthereader.Themainthemeisthestudyof
(i) theringZofintegers
(ii) theChineseRemainderTheoremandreciprocitylaws
(iii) (cid:2)nitegroupsfromthepointofviewofenumeration
(iv) abstractM(cid:246)biusInversion
(v) theroleofgeneratingfunctions
(vi) ringsofarithmeticfunctionsand
(vii) certainanaloguesoftheGoldbachproblem.
Manyinterestingtopicssuch as p-adic(cid:2)elds, cyclotomy,Emil Artin’s con-
jectureandFermat’sLastTheorem(FLT)havenotbeendiscussedindetail.How-
ever,theoverallpictureiswhatonegetsabouttheniceinterconnectionsbetween
numbertheoryandalgebra.
The monographhas been divided into four parts containing 16 chapters in
all.Eachchapterbeginswitha‘historicalperspective’andclosesbygiving‘notes
withillustrativeexamples/worked-outexample(s)’. PartIdealingwithelements
© 2007 by Taylor & Francis Group, LLC
ofnumbertheoryandalgebracontainssevenchapters. Thedetailsaregivenbe-
low.
PARTI
ELEMENTSOFNUMBERTHEORYANDALGEBRA
Chapter1:TheoremsofEuler,FermatandLagrange
Certainnewproofsofclassical theoremsofnumbertheoryarepointedout.
Using a counting principle of Melvin Hausner, the theorems of Fermat and
Lucas are proved. D. Zagier’s proofof Fermat’s two-squarestheorem is given.
Lagrange’s four-squares theorem is deduced from the fact that a certain 2 2
(cid:2)
matrixwithentriesfromZ[i]hasafactorizationofthetypeBB(cid:3) whenB(cid:3) isthe
adjoint (conjugate transpose of B). Linear Diophantine equations are also dis-
cussed.
Chapter2:Theintegraldomainofrationalintegers
Zisshownasanorderedintegraldomain. Itisprovedthatanorderedinte-
graldomainwhosesubsetofpositiveelementsiswell-ordered,isthesameasZ,
up to isomorphism. Operations on ideals of a commutative ring with unity are
described. Theygiveanaloguesof g.c.d. and l.c.m. of integers. In the case of
anintegraldomain,characterizationsofirreduciblesandprimesareshown. The
criterionforanintegraldomaintosatisfyUFDpropertyisgiven.Thenotionofa
GCDdomainisalsopointedout.
Chapter3:Euclideandomains
ZisaEuclideandomain.Theringofalgebraicintegersofaquadraticnumber
(cid:2)eld Q(pm) is a Euclidean domain when m=-1;-2;-3;-7and -11. ‘Almost
Euclidean’domainsarediscussed. ItisprovedthattheringR(-19)ofalgebraic
integersofQ(p-19)isaPID,butnotaEuclideandomain.Further,Zisshownto
betheuniqueEuclideandomainhaving‘double-remainderproperty’.
Chapter4:Ringsofpolynomialsandformalpowerseries
Polynomialringsareintroduced.IfFisa(cid:2)eld,theuniquenessofthedivision
algorithmin F[x] characterizesF[x] amongEuclideandomains. The ring of
A
arithmetic functions under the operations of addition and Dirichlet convolution
is shown to be a UFD via the ring C! of formalpower series (overthe (cid:2)eld C
ofcomplexnumbers)incountablyin(cid:2)niteindeterminates. Thissigni(cid:2)cantresult
is due to E. D. Cashwell and C.J.Everett. See ‘The ring of number-theoretic
functions’,Paci(cid:2)cJ.Math9(1959)975(cid:150)985.
Next, we give a formula for the number of monic irreducible polynomials
of degree m (>0) over the (cid:2)nite (cid:2)eld Z=pZ (where p is a prime) via M(cid:246)bius
inversion. It is deducedthat the numberof monicirreduciblepolynomialsover
Z=pZisin(cid:2)nite.
© 2007 by Taylor & Francis Group, LLC