资源描述：
《intro abstract algebra 1997 garrett》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、IntroAbstractAlgebrac1997-8,PaulGarrett,garrett@math.umn.eduhttp://www.math.umn.edu/~garrett/1Contents(1)BasicAlgebraofPolynomials(2)InductionandtheWell-orderingPrinciple(3)Sets(4)Somecountingprinciples(5)TheIntegers(6)Uniquefactorizationintoprimes(7)(*)PrimeNumbers(8)SunZe'sT
2、heorem(9)Goodalgorithmforexponentiation(10)Fermat'sLittleTheorem(11)Euler'sTheorem,PrimitiveRoots,Exponents,Roots(12)(*)Public-KeyCiphers(13)(*)PseudoprimesandPrimalityTests(14)Vectorsandmatrices(15)Motionsintwoandthreedimensions(16)PermutationsandSymmetricGroups(17)Groups:Lag
3、range'sTheorem,Euler'sTheorem(18)RingsandFields:de nitionsand rstexamples(19)Cyclotomicpolynomials(20)Primitiveroots(21)GroupHomomorphisms(22)CyclicGroups(23)(*)Carmichaelnumbersandwitnesses(24)Moreongroups(25)Finite elds(26)LinearCongruences(27)SystemsofLinearCongruences(28)A
4、bstractSunZeTheorem(29)(*)TheHamiltonianQuaternions(30)Moreaboutrings(31)Tables21.BasicAlgebraofPolynomialsCompletingthesquaretosolveaquadraticequationisperhapsthe rstreallygoodtrickinelemen-taryalgebra.Itdependsuponappreciatingtheformofthesquareofthebinomialx+y:22222(x+y)=x+x
5、y+yx+y=x+2xy+yThus,runningthisbackwards,aaaa22222x+ax=x+2()x=x+2()x+(),()2222aa22=(x+),()22Thenfora6=0,2ax+bx+c=0canberewrittenas0bcbcb2220==x+2x+=(x+)+,()a2aa2aa2aThus,bbc22(x+)=(),2a2aarbbc2x+=(),2a2aarbbc2x=,(),2a2aafromwhichtheusualQuadraticFormulaiseasilyobtained.Forpos
6、itiveintegersn,wehavethefactorialfunctionde ned:n!=123:::(n,2)(n,1)nAlso,wetake0!=1.Thefundamentalpropertyisthat(n+1)!=(n+1)n!Andthereistheseparatede nitionthat0!=1.Thelatterconventionhasthevirtuethatitworksoutinpractice,inthepatternsinwhichfactorialsaremostoftenused.Th
7、ebinomialcoecientsarenumberswithaspecialnotationn!n=kk!(n,k)!Thenamecomesfromthefactthatthesenumbersappearinthebinomialexpansion(expansionofpowersofthebinomial(x+y)):n(x+y)=nnnnnn,1n,222n,2n,1nx+xy+xy+:::+xy+xy+y12n,2n,13Xnn,ii=xyi0inNoticethatnn==1n0Therear
8、estandardidentitieswhichareusefulinanticipatingfactorizationo
