AsIseeit，thegraduatecourseinalgebramustprimarilypreparestudentstohandlethealgebrawhichtheywillmeetinallofmathematics：topology，partialdifferentialequations，differentialgeometry，algebraicgeometry，analysis，andrepresentationtheory，nottospeakofalgebraitselfandalgebraicnumbertheorywithallitsramifications.HenceIhaveinsertedthroughoutreferencestopapersandbookswhichhaveappearedduringthelastdecades，toindicatesomeofthedirectionsinwhichthealgebraicfoundationsprovidedbythisbookareused;Ihaveaccompaniedthesereferenceswithsomemotivatingcomments，toexplainhowthetopicsofthepresentbookfitintothemathematicsthatistocomesubsequentlyinvariousfields;andIhavealsomentionedsomeunsolvedproblemsofmathematicsinalgebraandnumbertheory.Theabcconjectureisperhapsthemostspectacularofthese.