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1、§1.ProblemofAlgebraLecture0Page1Lecture0INTRODUCTIONThislectureisanorientationonthecentralproblemsthatconcernus.Specifically,weidentifythreefamiliesof“FundamentalProblems”inalgorithmicalgebra(§1–§3).Intherestofthelecture(§4–§9),webrieflydiscussthecomplexity-theoreticbackground.§10collect
2、ssomecommonmathematicalterminologywhile§11introducescomputeralgebrasystems.Thereadermayprefertoskip§4-11onafirstreading,andonlyusethemasareference.Allourringswillcontainunitywhichisdenoted1(anddistinctfrom0).Theyarecommutativeexceptinthecaseofmatrixrings.Themainalgebraicstructuresofinte
3、restare:N=naturalnumbers0,1,2,...Z=integersQ=rationalnumbersR=realsC=complexnumbersR[X]=polynomialringind≥1variablesX=(X1,...,Xn)withcoefficientsfromaringR.LetRbeanyring.ForaunivariatepolynomialP∈R[X],weletdeg(P)andlead(P)denoteitsdegreeandleadingcoefficient(orleadingcoefficient).IfP=0thenby
4、definition,deg(P)=−∞andlead(P)=0;otherwisedeg(P)≥0andlead(P)=0.WesayPisa(respectively)integer,rational,realorcomplexpolynomial,dependingonwhetherRisZ,Q,RorC.Inthecourseofthisbook,wewillencounterotherrings:(e.g.,§I.1).Withtheexceptionofmatrixrings,allourringsarecommutative.Thebasicalgeb
5、raweassumecanbeobtainedfromclassicssuchasvanderWaerden[22]orZariski-Samuel[27,28].§1.FundamentalProblemofAlgebraConsideranintegerpolynomialniP(X)=aiX(ai∈Z,an=0).(1)i=0ManyoftheoldestproblemsinmathematicsstemfromattemptstosolvetheequationP(X)=0,(2)i.e.,tofindnumbersαsuchthatP(α)=0.Weca
6、llsuchanαasolutionofequation(2);alterna-tively,αisarootorzeroofthepolynomialP(X).Bydefinition,analgebraicnumberisazeroofsomepolynomialP∈Z[X].TheFundamentalTheoremofAlgebrastatesthateverynon-constantpoly-nomialP(X)∈C[X]hasarootα∈C.Putanotherway,Cisalgebraicallyclosed.d’Alembertfirstformul
7、atedthistheoremin1746butGaussgavethefirstcompleteproofinhis1799doctoralthesiscChee-KengYapMarch6,2000§1.ProblemofAlgebraLecture0Page2atHelmstedt.Itfollowsthattherearen(notnecessarilydistinct)complexnumbersα1,...,αn∈Csuchthatthepolynomialin(1)isequaltonP(X)≡an(X−αi).(3)i=1Toseethis,s
