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1、UndergraduateTextsinMathematicsEditorsS.AxlerF.W.GehringK.A.RibetAntoineChambert-LoirAFieldGuidetoAlgebraWith12IllustrationsAntoineChambert-LoirUniversite´deRennes1IRMAR,CampusdeBeaulieu35042RennesCedexFranceEditorialBoardS.AxlerF.W.GehringK.A.RibetMathematicsDepartmentMathematicsDepa
2、rtmentDepartmentofSanFranciscoStateEastHallMathematicsUniversityUniversityofMichiganUniversityofCaliforniaSanFrancisco,CA94132AnnArbor,MI48109atBerkeleyUSAUSABerkeley,CA94720-3840USAPicturedonthecover:Constructingthesquarerootofarealnumber(Pythagoras’stheorem).Seepage2fordiscussion.Ma
3、thematicsSubjectClassification(2000):12-01,12Fxx,11Rxx,13BxxLibraryofCongressCataloging-in-PublicationDataChambert-Loir,Antoine.Afieldguidetoalgebra/AntoineChambert-Loir.p.cm.—(Undergraduatetextsinmathematics,ISSN0172-6056)Includesbibliographicalreferencesandindex.ISBN0-387-21428-3(ha
4、rdback:alk.paper)1.Algebraicfields.I.Title.II.Series.QA247.C482004512′.3—dc222004048103ISBN0-387-21428-3Printedonacid-freepaper.©2005SpringerScience+BusinessMedia,Inc.Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerS
5、cience+BusinessMedia,Inc.,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnec-tionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinforma-tionstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevel
6、opedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.PrintedintheUnitedStatesofAmerica.(EBP)987654321SPIN10950579springeronlin
7、e.comThenthisisakindofknowledgewhichlegislationmayfitlyprescribe;andwemustendeavourtopersuadethosewhoaretobetheprincipalmenofourStatetogoandlearnarithmetic,notasamateurs,buttheymustcarryonthestudyuntiltheyseethenatureofnumberswiththemindonly;noragain,likemerchantsorretail-traders,witha
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