数论1－fermat的猜想与类域论

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1、TranslationsofM&THEMATICALMONOGRAPHSVolume186NumberTheory1Fermat’sDreamKazuyaKatoNobushigeKurokawaTakeshiSaitoTranslatedbyMasatoKuwatalallalllllllllllllllllllllllllllllllllllllllllllllIlllll~llllll~~lllFUDANBOO12090492443B'!$AmericanMathematicalSocietyProvidence,RhodeIlsandCont

2、entsPrefaceixPrefacetotheEnglishEditionxi...ObjectivesandOutlineoftheseBooksx111NotationxvChapter0.Introduction-FermatandNumberTheory~0.1.BeforeFermat0.2.Primenumbersandthesumoftwosquares0.3.p=x2+2y2,p=x2+3y2,...0.4.Pell’sequations0.5.Triangularnumbers,quadrangularnumbers,penta

3、gonalnumbers80.6.Triangularnumbers,squares,cubes100.7.Righttrianglesandellipticcurves110.8.Fermat’sLastTheorem12Exercises14Chapter1.RationalPointsonEllipticCurves171.1.Fermatandellipticcurves171.2.Groupstructureofanellipticcurve251.3.Mordell’stheorem30Summary43Exercises43Chapte

4、r2.ConiesandpadicNumbers452.1.Conies452.2.Congruence492.3.Coniesandquadraticresiduesymbols532.4.p-adicnumberfields58vii“IllCONTENTS2.5.Multiplicativestructureofthep-adicnumberfield692.6.Rationalpointsontonics74Summary78Exercises78Chapter3.<813.1.Threewondersofthevaluesofthe

5、ction813.2.Valuesatpositiveintegers843.3.Valuesatnegativeintegers89Summary99Exercises100Chapter4.AlgebraicNumberTheory1034.1.Methodofalgebraicnumbertheory1044.2.Theheartofalgebraicnumbertheory1134.3.Theclassnumberformulaforimaginaryquadraticfields1244.4.Fermat’sLastTheoremandKu

6、mmer127Summary132Exercises132AppendixA.RudimentsonDedekinddomains135A.l.DefinitionofaDedekinddomain135A.2.Fractionalideal136AnswerstoQuestions139AnswerstoExercises145Index153PrefaceThisbookwaswrittenin1996,twohundredyearsafter1796,whichwasaveryfruitfulyearforthegreatGauss,whoma

7、demanyfundamentalcontributionstomodernnumbertheory.Gausswasinhislateteensatthetime.OnMarch30hediscoveredamethodofconstructionofaregular17-gon.OnApril8heprovedthequadraticreciprocitylaw(see$2.2inthisvolume),whichhehimselfcalledagem.OnMay31heconjecturedwhatwouldlaterbecalled“thep

8、rimenumbertheorem”concerningthedistributionofprimenumb