Algebra and Number Theory. A Selection of Highlights

327 Pages·2017·1.506 MB·english

by  Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman

Benjamin Fine, Anthony Gaglione,
Anja Moldenhauer, Gerhard Rosenberger,
Dennis Spellman
Algebra and
Number Theory
A Selection of Highlights
MathematicsSubjectClassification2010
0001,00A06,1101,1201
Authors
Prof.Dr.BenjaminFine Prof.Dr.GerhardRosenberger
FairfieldUniversity UniversityofHamburg
DepartmentofMathematics DepartmentofMathematics
1073NorthBensonRoad Bundesstr.55
Fairfield,CT06430 20146Hamburg
USA Germany
Prof.Dr.AnthonyGaglione Prof.Dr.DennisSpellman
UnitedStatesNavalAcademy TempleUniversity
DepartmentofMathematics DepartmentofMathematics
212BlakeRoad 1801NBroadStreet
Annapolis,MD21401 Philadelphia,PA19122
USA USA
Dr.AnjaMoldenhauer
UniversityofHamburg
DepartmentofMathematics
Bundesstr.55
20146Hamburg
Germany
ISBN978-3-11-051584-8
e-ISBN(PDF)978-3-11-051614-2
e-ISBN(EPUB)978-3-11-051626-5
LibraryofCongressCataloging-in-PublicationData
ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2017WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
Coverimage:agsandrew/iStock/GettyImagesPlus
♾Printedonacid-freepaper
PrintedinGermany
www.degruyter.com
Preface
Tomanystudents,aswellastomanyteachers,mathematicsseemslikeamundane
discipline,filledwithrulesandalgorithmsanddevoidofbeautyandart.Howeverto
someonewhotrulydigsdeeplyintomathematicsthisisquitefarfromthetruth.The
worldofmathematicsispopulatedwithtruegems;resultsthatbothastoundandpoint
toaunityinboththeworldandaseeminglychaoticsubject.Itisoftenthatthesegems
andtheirsurprisingresultsareusedtopointtotheexistenceofaforcegoverningthe
universe;thatis,theypointtoahigherpower.Euler’smagicformula,eiπ+1=0,which
wegooverandproveinthisbookisoftencitedasaproofoftheexistenceofGod.While
tosomeoneseeingthisstatementforthefirsttimeitmightseemoutlandish,howeverif
onedelvesintohowthisresultisgeneratednaturallyfromsuchadisparatecollection
ofnumbersitdoesnotseemsostrangetoattributetoitacertainmysticalsignificance.
Unfortunatelymoststudentsofmathematicsonlyseebitsandpiecesofthisamaz-
ing discipline. In this book, which we call Algebra and Number Theory, we intro-
duceandexaminemanyoftheseexcitingresults.Weplannedthisbooktobeusedin
coursesforteachersandforthegeneralmathematicallyinterestedsoitissomewhat
betweenatextbookandjustacollectionofresults.Weexaminethesemathematical
gemsandalsotheirproofs,developingwhatevermathematicalresultsandtechniques
weneedalongtheway.InGermanyandtheUnitedStatesweseethebookasaMasters
LevelBookforprospectiveteachers.
WiththeincreasingdemandforeducationintheSTEMsubjects,thereisthereal-
izationthattogetbetterteachinginmathematics,theprospectiveteachersmustboth
bemoreknowledgeableinmathematicsandexcitedaboutthesubject.Thecoursesin
teacherpreparationdonottouchmanyoftheseresultsthatmakethedisciplinesoex-
citing.Thisbookisintendedtoaddressthisissue.ThefirstvolumeisonAlgebraand
NumberTheory.Wetouchonnumbersandnumbersystems,polynomialsandpoly-
nomialequations,geometryandgeometricconstructions.Thesepartsaresomewhat
independentsoaprofessorcanpickandchoosetheareastoconcentrateon.Much
morematerialisincludedthancanbecoveredinasinglecourse.Weproveallrele-
vantresultsthatarenottootechnicalorcomplicatedtoscarethestudents.Wefind
thatmathematicsisalsotiedtoitshistorysoweincludemanyhistoricalcomments.
Wetrytointroduceallthatisnecessaryhoweverwedopresupposecertainsub-
jects from school and undergraduate mathematics. These include basic knowledge
inalgebra,geometryandcalculusaswellassomeknowledgeofmatricesandlinear
equations.Beyondthesethebookisself-contained.
ThisfirstvolumeoftwoiscalledAlgebraandNumberTheory.Therearefourteen
chaptersandwethinkwehaveintroducedaverywidecollectionofresultsofthetype
thatwehavealludedtoabove.InChapters1–5welookathighlightsontheintegers.We
examineuniquefactorizationandmodulararithmeticandrelatedideas.Weshowhow
thesebecomecriticalcomponentsofmoderncryptographyespeciallypublickeycryp-
tographicmethodssuchasRSA.Threeoftheauthors(Fine,MoldenhauerandRosen-
berger)workpartlyascryptographerssocryptographyismentionedandexplained
inseveralplaces.InChapters4and5welookatexceptionalclassesofintegerssuch
astheFibonaccinumbersaswellastheFermatnumbers,Mersennenumbers,perfect
numbersandPythagoreantriples.Weexplainthegoldensectionaswellasexpressing
integersassumsofsquares.InChapters6–8welookatresultsinvolvingpolynomi-
alsandpolynomialequations.Weexplainfieldextensionsatanunderstandablelevel
andthenprovetheinsolvabilityofthequinticandbeyond.Theinsolvabilityofthe
quinticingeneralisoneoftheimportantresultsofmodernmathematics.
InChapters9–12welookathighlightsfromtherealandcomplexnumbersleading
eventuallytoanexplanationandproofoftheFundamentalTheoremofAlgebra.Along
thewayweconsidertheamazingpropertiesofthenumberseandπandproveindetail
thatthesetwonumbersaretranscendent.
Chapter13isconcernedwiththeclassicalproblemofgeometricconstructionsand
usesthematerialwedevelopedonfieldextensionstoprovetheimpossibilityofcertain
constructions.
FinallyinChapter14welookatEuclideanVectorSpaces.Wegiveseveralgeomet-
ricapplicationsandlookforinstanceatasecretsharingprotocolusingtheclosest
vectortheorem.
Wewouldliketothankthepeoplewhowereinvolvedinthepreparationofthe
manuscript.Theirdedicatedparticipationintranslatingandproofreadingaregrate-
fully acknowledged. In particular, we have to mention Anja Rosenberger, Annika
SchürenbergandthemanystudentswhohavetakentherespectivecoursesinDort-
mund,FairfieldandHamburg.Thosemathematical,stylistic,andorthographicerrors
thatundoubtedlyremainshallbechargedtotheauthors.Lastbutnotleast,wethank
deGruyterforpublishingourbook.
BenjaminFine
AnthonyGaglione
AnjaMoldenhauer
GerhardRosenberger
DennisSpellman
Contents
Preface|V
1 Thenatural,integralandrationalnumbers|1
1.1 Numbertheoryandaxiomaticsystems|1
1.2 Thenaturalnumbersandinduction|1
1.3 Theintegersℤ|10
1.4 Therationalnumbersℚ|13
1.5 Theabsolutevalueinℕ,ℤandℚ|15
2 Divisionandfactorizationintheintegers|19
2.1 TheFundamentalTheoremofArithmetic|19
2.2 Thedivisionalgorithmandthegreatestcommondivisor|23
2.3 TheEuclideanalgorithm|26
2.4 Leastcommonmultiples|30
2.5 Generalgcd’sandlcm’s|33
3 Modulararithmetic|39
3.1 Theringofintegersmodulon|39
3.2 UnitsandtheEulerφ-function|43
3.3 RSAcryptosystem|46
3.4 TheChineseRemainderTheorem|47
3.5 Quadraticresidues|54
4 Exceptionalnumbers|61
4.1 TheFibonaccinumbers|61
4.1.1 Thegoldenrectangle|67
4.1.2 Squaresinsemicircles|68
4.1.3 Sidelengthofaregular10-gon|69
4.1.4 Constructionofthegoldensectionαwithcompassandstraightedge
fromagivena∈ℝ,a>0|70
4.2 PerfectnumbersandMersennenumbers|71
4.3 Fermatnumbers|78
5 Pythagoreantriplesandsumsofsquares|83
5.1 ThePythagoreanTheorem|83
5.2 ClassificationofthePythagoreantriples|85
5.3 Sumofsquares|89
6 Polynomialsanduniquefactorization|95
6.1 Polynomialsoveraring|95
6.2 Divisibilityinrings|98
6.3 TheringofpolynomialsoverafieldK|100
6.3.1 Thedivisionalgorithmforpolynomials|101
6.3.2 Zerosofpolynomials|103
6.4 Horner-Scheme|108
6.5 TheEuclideanalgorithmandgreatestcommondivisorofpolynomials
overfields|112
6.5.1 TheEuclideanalgorithmforK[x]|114
6.5.2 UniquefactorizationofpolynomialsinK[x]|115
6.5.3 Generaluniquefactorizationdomains|116
6.6 PolynomialinterpolationandtheShamirsecretsharingscheme|117
6.6.1 Secretsharing|117
6.6.2 PolynomialinterpolationoverafieldK|117
6.6.3 TheShamirsecretsharingscheme|121
7 Fieldextensionsandsplittingfields|125
7.1 Fields,subfieldandcharacteristic|125
7.2 Fieldextensions|126
7.3 Finiteandalgebraicfieldextensions|131
7.3.1 Finitefields|134
7.4 Splittingfields|135
8 Permutationsandsymmetricpolynomials|141
8.1 Permutations|141
8.2 Cycledecompositionofapermutation|144
8.2.1 ConjugateelementsinSn|147
8.2.2 MarshallHall’sTheorem|148
8.3 Symmetricpolynomials|151
9 Realnumbers|157
9.1 Therealnumbersystem|157
9.2 Decimalrepresentationofrealnumbers|168
9.3 Periodicdecimalnumbersandtherationalnumber|172
9.4 Theuncountabilityofℝ|173
9.5 Continuedfractionrepresentationofrealnumbers|175
9.6 TheoremofDirichletandCauchy’sInequality|176
9.7 p-adicnumbers|178
9.7.1 NormedfieldsandCauchycompletions|179
9.7.2 Thep-adicfields|180
9.7.3 Thep-adicnorm|183
9.7.4 Theconstructionofℚp|184
9.7.5 Ostrowski’stheorem|185
10 Thecomplexnumbers,theFundamentalTheoremofAlgebraandpolynomial
equations|189
10.1 Thefieldℂofcomplexnumbers|189
10.2 Thecomplexplane|193
10.2.1 Geometricinterpretationofcomplexoperations|196
10.2.2 PolarformandEuler’sidentity|197
10.2.3 Otherconstructionsofℂ|201
10.2.4 TheGaussianintegers|201
10.3 TheFundamentalTheoremofAlgebra|202
10.3.1 FirstproofoftheFundamentalTheoremofAlgebra|204
10.3.2 SecondproofoftheFundamentalTheoremofAlgebra|207
10.4 Solvingpolynomialequationsintermsofradicals|209
10.5 SkewfieldextensionsofℂandFrobenius’sTheorem|220
11 QuadraticnumberfieldsandPell’sequation|227
11.1 Algebraicextensionsofℚ|227
11.2 Algebraicandtranscendentalnumbers|228
11.3 Discriminantandnorm|230
11.4 Algebraicintegers|235
11.4.1 Theringofalgebraicintegers|236
11.5 Integralbases|238
11.6 Quadraticfieldsandquadraticintegers|240
12 Transcendentalnumbersandthenumberseandπ|249
12.1 Thenumberseandπ|249
12.1.1 Calculationeofπ|251
12.2 Theirrationalityofeandπ|256
12.3 eandπthroughoutmathematics|263
12.3.1 Thenormaldistribution|263
12.3.2 TheGammaFunctionandStirling’sapproximation|264
12.3.3 TheWallisProductFormula|266
12.4 Existenceofatranscendentalnumber|270
12.5 Thetranscendenceofeandπ|273
12.6 Anamazingpropertyofπandaconnectiontoprimenumbers|282
13 Compassandstraightedgeconstructionsandtheclassical
problems|289
13.1 Historicalremarks|289
13.2 Geometricconstructions|289
13.3 Fourclassicalconstructionproblems|296
13.3.1 Squaringthecircle(problemofAnaxagoras500–428BC)|296
13.3.2 ThedoublingofthecubeortheproblemfromDeli|296
13.3.3 Thetrisectionofanangle|297
13.3.4 Constructionofaregularn-gon|298
14 Euclideanvectorspaces|303
14.1 Lengthandangle|303
14.2 OrthogonalityandApplicationsinℝ2andℝ3|309
14.3 Orthonormalizationandclosestvector|317
14.4 Polynomialapproximation|321
14.5 Secretsharingschemeusingtheclosestvectortheorem|323
Bibliography|327
Index|329
1 The natural, integral and rational numbers
1.1 Numbertheoryandaxiomaticsystems
Numbertheorybeginsasthestudyofthewholenumbersorcountingnumbers.For-
mallythecountingnumbers1,2,…arecalledthenaturalnumbersanddenotedbyℕ.
Ifweaddtothisthenumberzero,denotedby0,andthenegativewholenumberswe
getamorecomprehensivesystemcalledtheintegerswhichwedenotebyℤ.Thefocus
ofthisbookisonimportantandsometimessurprisingresultsinnumbertheoryand
thenfurtherresultsinalgebra.Manyresultsinnumbertheory,asweshallsee,seem
likemagic.Inordertorigorouslyprovetheseresultsweplacethewholetheoryinan
axiomaticsettingwhichwenowexplain.
Inmathematics,whendevelopingaconceptoratheoryitisoftennotpossible,all
usedterms,propertiesorclaimstoprove,especiallyexistenceofsomemathematical
fundamentals.Onecansolvethisproblemthenbyanaxiomaticapproach.Thebasis
ofatheorythenisasystemofaxioms:
– Certainobjectsandcertainpropertiesoftheseobjectsaretakenasgivenandac-
cepted.
– Aselectionofstatements(theaxioms)areconsideredbydefinitionastrueand
evident.
Atheoreminthetheorythenisatruestatement,whosetruthcanbeprovedfromthe
axiomswithhelpoftrueimplications.Asystemofaxiomsisconsistentifonecannot
proveastatementoftheform“AandnotA”.Theverificationisinindividualcases
oftenacomplicatedorevenanunsolvableproblem.Wearesatisfied,ifwecanquote
amodelforthesystemofaxioms,thatis,asystemofconcreteobjects,whichmeetall
thegivenaxioms.Asystemofaxiomsiscalledcategoricalifessentiallythereexists
only one model. By this we mean that for any two models we always get from one
modeltotheotherbyrenamingoftheobjects.Ifthisistruethenwehaveanaxiomatic
characterizationofthemodel.
Inthenextsectionweintroducethenaturalnumbersaxiomatically.
1.2 Thenaturalnumbersandinduction
ThenaturalnumbersℕarepresentedbythesystemofaxiomsdevelopedbyG.Peano
(1858–1932).Thisisdoneasfollows.
Thesetℕofthenaturalnumbersisdescribedbythefollowingaxioms:
(ℕ1) 1∈ℕ.
(ℕ2)Eacha∈ℕhasexactlyonesuccessora+∈ℕ.
(ℕ3)Alwaysisa+≠1,andforeachb≠1thereexistsana∈ℕwithb=a+.
(ℕ4)a≠b⇒a+≠b+.
2 | 1 Thenatural,integralandrationalnumbers
(ℕ5)IfT⊂ℕ,1∈T,andiftogetherwitha∈Talsoa+∈T,thenT=ℕ.
(Axiomofmathematicalinductionorjustinduction.)
Remarks1.1. (1) (ℕ2)and(ℕ4)meanthatthemap
σ∶ℕ→ℕ
a↦a+
isinjective.
(2) FromthePeanoaxiomswegetperdefinitionanaddition,amultiplicationandan
orderingforℕ:
(i) a+1∶=a+,
a+b+∶=(a+b)+,
(ii) a⋅1∶=a,
a⋅b+∶=ab+a,
(iii)a<b∶⇔∃x∈ℕwitha+x=b(“asmallerthanb”),
a≤b∶⇔a=bora<b(“aequalorsmallerthanb”).
Weneedtorecallsomedefinitions.
AsemigroupisasetH ≠∅togetherwithabinaryoperation⋅∶H ×H →H that
satisfiestheassociativepropertyforalla,b,c∈H:
(a⋅b)⋅c=a⋅(b⋅c).
Thesemigroupiscommutativeif
a⋅b=b⋅a.
Inthecommutativecaseweoftenwritetheoperationasaddition+insteadofmulti-
plication⋅.
AmonoidSisasemigroupwithaunityelemente,thatis,anelementewitha⋅e=
a=e⋅aforalla∈S;eisuniquelydetermined.
Moreover, a monoid S is called a group if for each a∈S there exists an inverse
elementa−1∈Swithaa−1=a−1a=e.Themonoidorgroupisnamedcommutativeor
abelianifinaddition
a⋅b=b⋅a foralla,b∈S.
Weoftenwrite1insteadofe.Wealsooftendrop⋅andusejustjuxtapositionforthis
operation.Ifweusetheaddition+weoftenwrite0insteadofeandcall0thezero
elementofS.
Theorem1.2. (1) Theadditionforℕisassociative,thatis,
a+(b+c)=(a+b)+c,