Numbers, polynomials, and games. An excursion into algebraic worlds PDF
Preview Numbers, polynomials, and games. An excursion into algebraic worlds
Numbers, Polynomials, and Games: An excursion into algebraic worlds John Perry August 24, 2016 Wonderisthedesiretounderstandanobservationwhosecauseeludesus or exceeds our knowledge. So wonder can stimulate pleasure, insofar as itstimulatesahopeofunderstandingwhatweobserve. Thisiswhywon- drousthingspleaseus. —ThomasAquinas,SummaTeologica,Primaparssecundæpar- tis,q.32art. 8co. (loosetranslation) Copyright2015byJohnPerry TypesetusingLyxandLATEX,intheGentiumtypeface,copyrightSILinternational. See www.lyx.org,www.tug.org,www.sil.orgfordetails. SomequoteswerefoundusingtheMathematicalQuotationServeratFurmanUniver- sity. Contents Preface viii 1 Noetherianbehavior 1 1¨1 Twogames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Nim IdealNim 1¨2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Fundamentalsets Setarithmetic 1¨3 Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Partialorderings Linearorderings 1¨4 Wellorderinganddivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Wellordering Division TheequivalenceoftheWell-OrderingPrincipleandInduction 1¨5 Divisiononthelattice(optional) . . . . . . . . . . . . . . . . . . . . . . . . . 29 1¨6 Polynomialdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Algebraicsystemsandstructures 37 2¨1 Fromsymmetrytoarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 “Nimbers” Nimberequivalence Nimberaddition WhataboutIdealNim? Self-cancelingarithmetic Clockworkarithmeticofintegers 2¨2 Propertiesandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Propertieswithoneoperation Sodoesadditionofremaindersformamonoid,orevenagroup? Whataboutstructureswithtwooperations? Cayleytables 2¨3 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Theidea Thedefinition i CONTENTS ii Sometimes,lessismore DirectProducts 3 Commonandimportantalgebraicsystems 69 3¨1 Polynomials,realandcomplexnumbers . . . . . . . . . . . . . . . . . . . . . 69 Polynomialremainders Realnumbers Complexnumbers 3¨2 Therootsofunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Ageometricpattern Agroup! 3¨3 Cyclicgroups;theorderofanelement . . . . . . . . . . . . . . . . . . . . . . 83 Exponents Cyclicgroupsandgenerators Theorderofanelement 3¨4 Anintroductiontofiniteringsandfields . . . . . . . . . . . . . . . . . . . . . 90 Characteristicsoffiniterings Evaluatingpositionsinthegame 3¨5 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Matrixarithmetic Propertiesofmatrixarithmetic 3¨6 Symmetryinpolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 IntuitivedevelopmentofD 3 DetailedproofthatD containsallsymmetriesofthetriangle 3 4 SubgroupsandIdeals,CosetsandQuotients 118 4¨1 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4¨2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Definitionandexamples Importantpropertiesofideals 4¨3 Thebasisofanideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Idealsgeneratedbymorethanoneelement Principalidealdomains 4¨4 Equivalencerelationsandclasses . . . . . . . . . . . . . . . . . . . . . . . . . 138 4¨5 Clockworkringsandideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4¨6 Partitioninggroupsandrings . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Theidea PropertiesofCosets 4¨7 Lagrange’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4¨8 QuotientRingsandGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Quotientrings “Normal”subgroups Quotientgroups Conjugation 4¨9 TheIsomorphismTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 CONTENTS iii Motivatingexample TheIsomorphismTheorem 5 Numbertheory 179 5¨1 TheEuclideanAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Commondivisors TheEuclideanAlgorithm TheEuclideanAlgorithmandBezout’sLemma 5¨2 Acardtrick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 ThesimpleChineseRemainderTheorem AgeneralizedChineseRemainderTheorem 5¨3 TheFundamentalTheoremofArithmetic . . . . . . . . . . . . . . . . . . . . 193 5¨4 Multiplicativeclockworkgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Clockworkmultiplication Amultiplicativeclockworkgroup 5¨5 Euler’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Computingφpnq Fastexponentiation 5¨6 RSAEncryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Descriptionandexample Theory Sageprograms Mapleprograms 6 Factorization 213 6¨1 Awrinklein“prime” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Primeandirreducible: adistinction Primeandirreducible: adifference 6¨2 Theidealsoffactoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Idealsofirreducibleandprimeelements Howareprimeandirreducibleelementsrelated? 6¨3 Timetoexpandourdomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Uniquefactorizationdomains Euclideandomains 6¨4 Fieldextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Extendingaring Extendingafieldtoincludearoot 6¨5 FiniteFieldsI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Quickreview Buildingfinitefields 6¨6 FinitefieldsII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Theexistenceoffinitefields Euler’stheorems 6¨7 Polynomialfactorizationinfinitefields . . . . . . . . . . . . . . . . . . . . . . 246 Distinctdegreefactorization. CONTENTS iv Equaldegreefactorization Squarefreefactorizationoverafieldofnonzerocharacteristic 6¨8 Factoringintegerpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Squarefreefactorizationoverafieldofcharacteristiczero Onebigirreducible. Severalsmallprimes. 7 Someimportant,noncommutativegroupsandrings 258 7¨1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Additionandmultiplicationoffunctions Functionsundercomposition Differentiationandintegration 7¨2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Groupsofpermutations Ahintofthingstocome. 7¨3 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Homomorphisms Isomorphisms 8 Groupsofpermutations 274 8¨1 Cyclenotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Cycles Cyclearithmetic Permutationsascycles 8¨2 Cayley’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 8¨3 Alternatinggroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Transpositions Evenandoddpermutations Thealternatinggroups 8¨4 The15-puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9 Solvingpolynomialsbyradicals 296 9¨1 Radicalextensionsofafield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Extendingafieldbyaroot Complexroots 9¨2 Thesymmetriesoftherootsofapolynomial . . . . . . . . . . . . . . . . . . . 303 9¨3 Galoisgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Isomorphismsoffieldextensionsthatpermutetheroots Solvingpolynomialsbyradicals 9¨4 “Solvable”groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 9¨5 TheTheoremofAbelandRuffini . . . . . . . . . . . . . . . . . . . . . . . . . 316 A“reverse-Lagrange”Theorem Wecannotsolvethequinticbyradicals 9¨6 TheFundamentalTheoremofAlgebra . . . . . . . . . . . . . . . . . . . . . . 324 BackgroundfromCalculus CONTENTS v Somemorealgebra ProofoftheFundamentalTheorem 10 Rootsofpolynomialsystems 328 10¨1 Gaussianelimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 10¨2 Monomialorderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Thelexicographicordering Monomialdiagrams Thegradedreverselexicographicordering Admissibleorderings 10¨3 Atriangularformforpolynomialsystems . . . . . . . . . . . . . . . . . . . . 343 Amatrixpointofview Anidealpointofview Buchberger’salgorithm 10¨4 Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 10¨5 Elementaryapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 AGro¨bnerbasisofanideal AGro¨bnerbasisandavariety Nomenclature rrs theelementr`nZofZ n (cid:104)g(cid:105) thegroup(orideal)generatedbyg я theidentityelementofamonoidorgroup }P} thesqaredistanceofthepointPtotheorigin sq a ” b aisequivalenttob(modulod) d A thealternatinggrouponthreeelements 3 AŸG forGagroup,AisanormalsubgroupofG AŸR forRaring,AisanidealofR AutpSq thegroupofautomorphismsonS rG,Gs commutatorsubgroupofagroupG rx,ys forxandyinagroupG,thecommutatorofxandy D pRq thesetofalldiagonalmatriceswhosevaluesalongthediagonalisconstant n dZ thesetofintegermultiplesofd Fpαq fieldextensionofFbyalpha G{A thesetofleftcosetsofA GzA thesetofrightcosetsofA gA theleftcosetofAwithg GL pRq thegenerallineargroupofinvertiblematrices m gz forGagroupandg,z P G,theconjugationofgbyz,orzgz´1 H ă G forGagroup,HisasubgroupofG lcmpt,uq theleastcommonmultipleofthemonomialstandu vi CONTENTS vii lmppq theleadingmonomialofthepolynomialp lvppq theleadingvariableofalinearpolynomialp N2 thetwo-dimensionallatticeofnaturalnumbers,onwhichweplayIdealNim. N pHq thenormalizerofasubgroupHofG G Ω thenthrootsofunity;thatis,allrootsofthepolynomialxn ´1 n ordpxq theorderofx PpSq thepowersetofS Q thegroupofquaternions 8 (cid:104)r ,r ,. . . ,r (cid:105) theidealgeneratedbyr ,r ,. . . ,r 1 2 m 1 2 m R thesetofrealnumbers,orallpossibledistancesonecanmovealongaline S thegroupofallpermutationsofalistofnelements n sqdpP,Qq thesqaredistancebetweenthepointsPandQ ω typically,aprimitiverootofunity X thesetofmonomials,ineitheroneormanyvariables(thelattersometimesasX n ZpGq centralizerofagroupG Zris theGaussianintegers,a`bi : a,b P Z Z˚ thesetofelementsofZ thatarenotzerodivisors n n Preface Awisemanspeaksbecausehehassomethingtosay; afoolbecausehehastosay something. —Plato Why this text? Thistexthasthreegoals. The first goal is to introduce you to the algebraic view of the world. This view reveals strangemathematicalcreaturesthatconnectseeminglyunrelatedmathematicalideas. Ihave triedtoorganizetheexcursionsothat,bythetimeyou’redonereadingatleastthefirstchap- terortwo,youwillunderstandthattheworldweinhabitisnotmerelydifferent,butwonder- fullydifferent. Thesecondgoalistotakeyouimmediatelyintothiswonderfulworld. Whileitispossible toteachalgebrawithoutevermentioningpolynomials,andthatisinfacthowIlearnedit,a student can find himself left with a gnawing question: What do groups, rings, etc. have to dowith“algebra”? Surelytheyholdsomerelationshiptopolynomialsandsolvingequations? Thealgebraicworldstrikesthenewcomerasexotic,butthere’snoreasonithastobeesoteric. Youwillencounterpolynomialsandtheirrootsintheveryfirstchapter—indeed,inthevery firstpages,thoughhowtheyappearwon’tbeclearuntillater. Thethirdgoalistoleadyouonanintuitivepathintothisworld. Higheralgebraisoften called“abstract”algebra,withreason. Abstractionisdifficult,andrequiresacertainamount ofmaturity,patience,andperseverance. Proofsareabigpartofalgebra,butmanystudents arrive in the course with no more experience than a survey on proof techniques. One class on proofs does not a proof-writer make! Reflecting on my own experience as a student: I reachedtherequisitematuritylaterthanmanyofmyfellowstudents. Thisinitiallydeterred me from pursuing doctoral studies, and even then it took me a while. I like to tell students thatIdon’thaveaPhDbecauseI’msmart;IhaveaPhDbecauseIwastoodumbtoquit. There’s truthtothat,butIwasalsoluckytohavehadtwograduateprofessorswhospentalotoftime elaboratingonbothhowtofindjustificationforanidea,andhowtowritetheproof. Itryto dothatinclassmyself,andmanyexercisesprovidehintsonhowtobeginandwheretolook. What should you do? Algebraisprobablydifferentfromthemathclassesyou’vehadbefore. Ratherthancomputa- tion,itexpectsexplanation. viii
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