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1、AbstractAlgebraTheoryandApplicationsThomasW.JudsonStephenF.AustinStateUniversityFebruary14,2009iic1997byThomasW.Judson.Permissionisgrantedtocopy,distributeand/ormodifythisdocumentun-derthetermsoftheGNUFreeDocumentationLicense,Version1.2oranylaterversionpublishedbytheFreeSoftwareFoundation;withnoInva
2、ri-antSections,noFront-CoverTexts,andnoBack-CoverTexts.AcopyofthelicenseisincludedintheappendixentitledGNUFreeDocumentationLicense".Acurrentversionofthistextcanalwaysbefoundatabstract.ups.edu.PrefaceThistextisintendedforaone-ortwo-semesterundergraduatecourseinabstractalgebra.Traditionally,thesecour
3、seshavecoveredthetheoreti-calaspectsofgroups,rings,and elds.However,withthedevelopmentofcomputinginthelastseveraldecades,applicationsthatinvolveabstractal-gebraanddiscretemathematicshavebecomeincreasinglyimportant,andmanyscience,engineering,andcomputersciencestudentsarenowelectingtominorinmathematic
4、s.Thoughtheorystilloccupiesacentralroleinthesubjectofabstractalgebraandnostudentshouldgothroughsuchacoursewithoutagoodnotionofwhataproofis,theimportanceofapplicationssuchascodingtheoryandcryptographyhasgrownsigni cantly.Untilrecentlymostabstractalgebratextsincludedfewifanyapplica-tions.However,oneof
5、themajorproblemsinteachinganabstractalgebracourseisthatformanystudentsitistheir rstencounterwithanenviron-mentthatrequiresthemtodorigorousproofs.Suchstudentsoften ndithardtoseetheuseoflearningtoprovetheoremsandpropositions;appliedexampleshelptheinstructorprovidemotivation.Thistextcontainsmoremateria
6、lthancanpossiblybecoveredinasinglesemester.Certainlythereisadequatematerialforatwo-semestercourse,andperhapsmore;however,foraone-semestercourseitwouldbequiteeasytoomitselectedchaptersandstillhaveausefultext.Theorderofpresen-tationoftopicsisstandard:groups,thenrings,and nally elds.Emphasiscanbeplaced
7、eitherontheoryoronapplications.Atypicalone-semestercoursemightcovergroupsandringswhilebrie
ytouchingon eldtheory,usingChapters0through5,8,9,11(the rstpart),14,15,16(the rstpart),18,and19.Partsofthesec
