Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

394 Pages·2019·2.46 MB·english

by  J L Lehman

AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 52
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Quadratic Number Theory is an introduction to algebraic number t]heory
for readers with a moderate knowledge of elementary number theory and Q
some familiarity with the terminology of abstract algebra. By restricting QUADRATIC
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attention to questions about squares the author achieves the dual goals of 2 3
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making the presentation accessible to undergraduates and refl ecting the
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historical roots of the subject. The representation of integers by quadratic
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forms is emphasized throughout the text. a NUM]BE R
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[ Lehman introduces an innovative notation for ideals of a quadratic domain ti 3:
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notation , serv[e the author’s historical purpose as well; ideals can be seen . THEORY An Invitation
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as number-like objects,3 as Kummer and Dedekind conceived of them. The e
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notation can be adapted to quadratic forms an1d provides insight into the m
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connection between quadratic forms and ideals. The computation of class n b Methods in the
groups and continued fraction representations are featured—the author’s e
J. L. Lehman Higher Arithmetic
notation makes these computations particularly illuminating. r

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Quadratic Number Theory, with its exceptionally clear prose, hundreds of
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AMS/MAA DOLCIANIMATHEMATICALEXPOSITIONS
VOL 52
Quadratic
Number Theory
An Invitation to Algebraic Methods
in the Higher Arithmetic
J. L. Lehman
DolcianiMathematicalExpositionsEditorialBoard
HarrietS.Pollatsek,Editor
PriscillaS.Bremser ThomasA.Richmond
AlfredM.Dahma C.RayRosentrater
ElizabethDenne AyseA.Sahin
EmilyH.Moore DanE.Steffy
KatharineOtt
2010MathematicsSubjectClassification.Primary11R11,
11R29,11R27,11E25,11E16,11A55,11B50,11Y40.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/dol-52
LibraryofCongressCataloging-in-PublicationData
Names:Lehman,J.L.(JamesLarry),1957–author.
Title: Quadraticnumbertheory: Aninvitationtoalgebraicmethodsinthehigherarithmetic/J.L.
Lehman.
Description:Providence,RhodeIsland:MAAPress,animprintoftheAmericanMathematicalSoci-
ety,[2019]|Series:Dolcianimathematicalexpositions;volume52|Includesanindex.
Identifiers:LCCN2018040720|ISBN9781470447373(alk.paper)
Subjects: LCSH:Algebraicnumbertheory. |Numbertheory. |Quadraticfields. |Algebraicfields.
|AMS:Numbertheory–Algebraicnumbertheory: globalfields–Quadraticextensions. msc|
Numbertheory–Algebraicnumbertheory: globalfields–Classnumbers,classgroups,discrim-
inants. msc|Numbertheory–Algebraicnumbertheory: globalfields–Unitsandfactorization.
msc|Numbertheory–Formsandlinearalgebraicgroups–Sumsofsquaresandrepresentations
byotherparticularquadraticforms. msc|Numbertheory–Formsandlinearalgebraicgroups–
Generalbinaryquadraticforms. msc|Numbertheory–Elementarynumbertheory–Continued
fractions.msc|Numbertheory–Sequencesandsets–Sequences(mod𝑚).msc|Numbertheory
–Computationalnumbertheory–Algebraicnumbertheorycomputations.msc
Classification:LCCQA247.L419552019|DDC512.7/4–dc23
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Contents
Preface vii
Acknowledgments xiii
Introduction: ABriefReviewofElementaryNumberTheory 1
0.1 LinearEquationsandCongruences 1
0.2 QuadraticCongruencesModuloPrimes 6
0.3 QuadraticCongruencesModuloCompositeIntegers 10
PartOne: QuadraticDomainsandIdeals 15
1 GaussianIntegersandSumsofTwoSquares 17
1.1 SumsofTwoSquares 17
1.2 GaussianIntegers 25
1.3 IdealFormforGaussianIntegers 32
1.4 FactorizationandMultiplicationwithIdealForms 38
1.5 ReductionofIdealFormsforGaussianIntegers 44
1.6 SumsofTwoSquaresRevisited 48
GaussianIntegersandSumsofTwoSquares—Review 52
2 QuadraticDomains 55
2.1 QuadraticNumbersandQuadraticIntegers 56
2.2 DomainsofQuadraticIntegers 61
2.3 IdealFormforQuadraticIntegers 68
2.4 IdealNumbers 74
2.5 QuadraticDomainswithUniqueFactorization 80
2.6 QuadraticDomainswithoutUniqueFactorization 86
QuadraticDomains—Review 91
3 IdealsofQuadraticDomains 93
3.1 IdealsandIdealNumbers 94
3.2 WritingIdealsasIdealNumbers 99
3.3 PrimeIdealsofQuadraticDomains 103
3.4 MultiplicationofIdeals 107
iii
iv Contents
3.5 PrimeIdealFactorization 112
3.6 AFormulaforIdealMultiplication 119
IdealsofQuadraticDomains—Review 126
PartTwo: QuadraticFormsandIdeals 129
4 QuadraticForms 131
4.1 ClassificationofQuadraticForms 131
4.2 EquivalenceofQuadraticForms 136
4.3 RepresentationsofIntegersbyQuadraticForms 140
4.4 GeneraofQuadraticForms 145
QuadraticForms—Review 151
5 CorrespondencebetweenFormsandIdeals 153
5.1 EquivalenceofIdeals 154
5.2 QuadraticFormsAssociatedtoanIdeal 158
5.3 CompositionofBinaryQuadraticForms 164
5.4 ClassGroupsofIdealsandQuadraticForms 170
CorrespondencebetweenFormsandIdeals—Review 175
PartThree: PositiveDefiniteQuadraticForms 177
6 ClassGroupsofNegativeDiscriminant 179
6.1 ReducedPositiveDefiniteQuadraticForms 179
6.2 CalculationofIdealClassGroups 186
6.3 GeneraofIdealClasses 190
ClassGroupsofNegativeDiscriminant—Review 197
7 RepresentationsbyPositiveDefiniteForms 199
7.1 NegativeDiscriminantswithTrivialClassGroups 200
7.2 PrincipalSquareDomains 205
7.3 QuadraticDomainsthatAreNotPrincipalSquareDomains 212
7.4 ConstructionofRepresentations 219
RepresentationsbyPositiveDefiniteForms—Review 225
8 ClassGroupsofQuadraticSubdomains 227
8.1 ConstructingClassGroupsofSubdomains 227
8.2 ProjectionHomomorphisms 234
8.3 TheKernelofaProjectionHomomorphism 239
ClassGroupsofQuadraticSubdomains—Review 244
PartFour: IndefiniteQuadraticForms 245
9 ContinuedFractions 247
9.1 IntroductiontoContinuedFractions 247
9.2 Pell’sEquation 253
Contents v
9.3 ConvergenceofContinuedFractions 259
9.4 ContinuedFractionExpansionsofRealNumbers 265
9.5 PurelyPeriodicContinuedFractions 269
9.6 ContinuedFractionsofIrrationalQuadraticNumbers 274
ContinuedFractions—Review 281
10 ClassGroupsofPositiveDiscriminant 285
10.1 ClassGroupsofIndefiniteQuadraticForms 285
10.2 GeneraofQuadraticFormsandIdeals 291
10.3 ContinuedFractionsofIrrationalQuadraticNumbers 296
10.4 EquivalenceofIndefiniteQuadraticForms 301
ClassGroupsofPositiveDiscriminant—Review 304
11 RepresentationsbyIndefiniteForms 307
11.1 TheContinuedFractionofaQuadraticForm 307
11.2 UnitsandAutomorphs 316
11.3 ExistenceofRepresentationsbyIndefiniteForms 320
11.4 ConstructingRepresentationsbyIndefiniteForms 326
RepresentationsbyIndefiniteForms—Review 331
PartFive: QuadraticRecursiveSequences 333
12 PropertiesofRecursiveSequences 335
12.1 DivisibilityPropertiesofQuadraticRecursiveSequences 336
12.2 PeriodicityofQuadraticRecursiveSequences 343
12.3 SuborderFunctions 349
12.4 SuborderSequences 356
PropertiesofRecursiveSequences—Review 364
13 ApplicationsofQuadraticRecursiveSequences 367
13.1 RecursiveSequencesandAutomorphs 367
13.2 AnApplicationtoPell’sEquation 371
13.3 QuadraticSubdomainsofPositiveDiscriminant 376
ApplicationsofQuadraticRecursiveSequences—Review 381
ConcludingRemarks 383
ReferencesandSuggestedReading 384
ListofNotation 387
Index 391

Preface
This book is intended as an introduction to algebraic methods in number the-
ory,suitableformathematicsstudentsandotherswithamoderatebackground
inelementarynumbertheoryandtheterminologyofabstractalgebra. Although
oftenfirstencounteredatthegraduatelevel, algebraicnumbertheorycanbea
valuablefieldofstudyforundergraduatemathematicsstudents,providingcon-
textforandconnectionsbetweendifferentareasofthemathematicscurriculum
andservingasamotivationforthehistoricaldevelopmentofabstractalgebraic
concepts. Wehaveaimedthistexttowardundergraduatestudentsandnonspe-
cialistsbyrestrictingourattentiontoquestionsarisingfromsquaresofintegers,
thus referring to our topic as quadraticnumbertheory. This grounding in eas-
ily stated problems motivates the key concepts of algebraic number theory but
allowsfor“hands-on”computationaltechniques,oftenasapplicationsoftopics
fromelementarynumbertheory. Manyofthesemethodsareapproachedinan
originalwayinthistext,whichwedescribefurtherinthispreface.
Background. Numbertheory(sometimescalledthehigherarithmetic)isde-
finedbroadlyasthestudyofthepropertiesofintegers. Manyarithmeticprob-
lemscanbeapproachedby“elementary”methods,thatis,intermsofthesetof
integersitself. Butinsomecases,propertiesofintegersmightbemosteasilyob-
tainedandunderstoodbyworkingwithinlargersetsofnumbers. Anexample,
which we will take as our starting point in Chapter 1, is the classical problem
of determining which integers can be written as a sum of two squares. While
we can answer this question by elementary means, as we will see in §1.1, the
resultscanbemorenaturallyexplainedbyappealingtothesetofGaussianinte-
gers,ℤ[𝑖]={𝑞+𝑟𝑖 ∣𝑞,𝑟 ∈ℤ},where𝑖2 =−1. Aninteger𝑛thatisasumoftwo
squaresisalsoaproductoftwoGaussianintegers:
𝑛=𝑞2+𝑟2 =(𝑞+𝑟𝑖)(𝑞−𝑟𝑖).
Writingthesumasaproductallowsustorephrasetheproblemintermsoffactor-
izationof𝑛intheGaussianintegers,whereaclassificationofirreducibleelements
ofℤ[𝑖]leadstoacompletedescriptionofsumsoftwosquaresofintegers.
vii
viii Preface
Similarexamples,suchasrepresentationsofintegersas𝑥2+2𝑦2or𝑥2+3𝑦2,
mightbeapproachedusingnumbersoftheform𝑞+𝑟√−2or𝑞+𝑟√−3. Intro-
ducing these numbers raises new questions, however. Which numbers of this
typearemostanalogoustointegersintheirproperties? Whichelementsinthese
sets can be regarded as “prime” factors, and how can we distinguish between
differentprimes? Doesuniquenessofprimefactorization—afamiliarproperty
of the integers—hold in these more general sets of numbers? For instance, in
ℤ[√−7]={𝑞+𝑟√−7∣𝑞,𝑟 ∈ℤ},theequation
2⋅2⋅2=8=(1+√−7)(1−√−7)
appearstopresenttwodifferentfactorizationsofanumberintotermsthatcannot
bebrokendownfurther. Inthisexample,forreasonswewillclarifylater,itturns
outthatabettersettingforthisfactorizationistheset
1+√−7
𝐷 ={𝑞+𝑟𝑧||𝑞,𝑟 ∈ℤand𝑧= }.
−7 2
1−√−7
If𝑧= =1−𝑧,wefindthat
2
2⋅2⋅2=(𝑧⋅𝑧)(𝑧⋅𝑧)(𝑧⋅𝑧)=(𝑧2⋅𝑧)(𝑧⋅𝑧2)=(1+√−7)(1−√−7),
sothattheapparentlydifferentfactorizationsaremerelydifferentgroupingsof
thesameterms. Asanotherexample,thefollowingmightappeartopresentdif-
ferentfactorizationsof7inℤ[√2]={𝑞+𝑟√2∣𝑞,𝑟 ∈ℤ}:
(3+√2)(3−√2)=7=(5+4√2)(−5+4√2).
Thistime,however,wefindthat
(5+4√2)(−5+4√2)=(3+√2)(1+√2)(−1+√2)(3−√2)=(3+√2)(3−√2),
andwesaythatthefactorsareunitmultiplesofeachother,orassociates,soare
notregardedasdistinctfactorizations(inthesamewaythat3⋅5and(−3)(−5)
arenotviewedasdifferentwaysoffactoring15intheintegers). Butwewilllater
showthatotherexamplesofdistinctfactorizations,suchas
2⋅3=6=(1+√−5)(1−√−5)
inthesetℤ[√−5]={𝑞+𝑟√−5∣𝑞,𝑟 ∈ℤ},cannotberesolvedineitherofthese
ways.
ErnstKummer(1810–1893)madeamajoradvanceinattackingthisproblem
of distinct irreducible factorization, reasoning that in such cases, the apparent
lackofuniquenesscouldberemediedbyconsideringalargersetof“idealnum-
bers,” in which terms factor further, similarly to the case of ℤ[√−7] and 𝐷
−7
notedabove. Asanoften-usedanalogytoKummer’sconcept,supposethatsome
alienbeingsknowallaboutevenintegers,buthavenoconceptofoddintegers.
Preface ix
Theset𝐸ofevenintegershasmanypropertiesincommonwithoursetℤ(aside
from an identity element for multiplication). In particular, we can generalize
theconceptofdivisibilityin𝐸,wherewecouldsaythat𝑎divides𝑏ifthereisan
element𝑞in𝐸suchthat𝑏 =𝑎𝑞. Forinstance,wewouldsaythat6divides12be-
cause12=6⋅2with2in𝐸,butthat6doesnotdivide30becausethereisnoeven
integerthatwecanmultiplyby6toobtain30. Wemightthendefineanelement
𝑝in𝐸tobeprimeif𝑝cannotbewrittenasaproductoftwoelementsin𝐸. The
primeelementsof𝐸 arepreciselythenumbersthatare(inourusualterminol-
ogy)congruentto2modulo4,thatis,2,6,10,14,andsoforth.Wefindthatevery
elementof𝐸 canbewrittenasaproductofprimes. Butthisfactorizationisnot
alwaysunique, asillustratedforexampleby60, whichcanbewritteneitheras
2⋅30oras6⋅10. Ofcourse,knowingaboutoddintegers,werecognizethesetwo
factorizationsasdifferentcombinationsofthe“true”primefactorizationof60:
2⋅30=2⋅(2⋅3⋅5)=(2⋅3)⋅(2⋅5)=6⋅10.
However,thealiensfamiliaronlywithevenintegersmightregardtheseoddin-
tegersasidealnumbers,introducedsimplytoobtainuniquefactorization.
Kummerdidnotdefineidealnumbersprecisely, butratherdescribedonly
theirdivisibilityproperties.RichardDedekind(1831–1916)recognizedthatideal
numberscouldbedefinedascertaintypesofsubsets, whichhecalled“ideals,”
of a set such as ℤ[√−5], and so be studied in a concrete way. In addition to
clarifyingKummer’swork,Dedekind’sdefinitionofidealsfoundapplicationsto
otheralgebraicproblemsandwasamajorfactorinthedevelopmentofmodern
abstractalgebrainthenineteenthandtwentiethcenturies.
Thereisnodoubt,however,thatDedekindregardedidealsasbeing“num-
bers,” in some sense, as had Kummer. In his 1877 treatise TheoryofAlgebraic
Integers,Dedekindcomparedthedefinitionofidealstohisearlierdevelopment
of irrational numbers as certain types of subsets of the rational numbers, now
known as Dedekind cuts, which placed the set of real numbers on firm logical
ground. Thustheconceptofanumberbeingdefinedasasetwasnotunnatural
forhim.
Onegoalofthisbookistorecapturethis“numerical”interpretationofideals.
Wecandoso,aswedescribeinthenextsubsection,inthespecialcaseofdomains
ofquadraticintegersdefinedintermsofrootsofdegreetwopolynomialswithin-
tegercoefficients. (TheworkofKummerandDedekindwasinabroadersetting
ofalgebraicintegers,usingrootsofpolynomialsofarbitrarydegreehavinginteger
coefficients.) Thuswemayregardquadraticnumbertheory,andthistextinpar-
ticular,asasteppingstonetowardtheconceptsandmethodsofalgebraicnumber
theory.
Innovative Aspects of the Text. Thisbookisdividedintofiveparts,each
consistingoftwoorthreechapters. Eachparthasaseparateintroduction,and