数论， An introduction to the theory of numbers, Niven, Montgomery, 第五版

图书标准信息

• 作者 [德]哈塞 著
• 出版社 世界图书出版公司
• 出版时间 2010-09
• 版次 1
• ISBN 9787510027352
• 定价 79.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 638页

【内容简介】

Inspiteofthefactthatnowadaystherearequiteafewbooksonalgebraicnumbertheoryavailabletothemathematicalcommunity,thereseemstobestillastrongneedforafundamentalworklikeIIasse's,,Zahlentheorie".ThisimpressioniscorroboratedbythegreatnumberofinquiriestheeditorreceivedaboutthedateofappearanceoftheEnglishtranslationofHasse'sbook.Onemainreasonfortheunbrokeninterestinthisbookliesprobablyinitsvividpresentationofthedivisortheoreticapproachtoalgebraicnumbertheory,anapproachwhichwasdevelopedbyHasse'sformerteacherIIenselandfurtherexpandedbyHassehimseff.Hassedoesnotcontenthimselfwithamerepresentationofthenumber-theoreticmaterial,buthemotivatesthebasicideasandquestions,commentsonthemindetail,andpointsouttheirconnectionswithneighboringbranchesofmathematics.

【目录】

partⅰ.thefoundationsofarithmeticintherationalnumberfield
chapter1.primedecomposition
functionfields
chapter2.divisibility
functionfields
chapter3.congruences
functionfields
thetheoryoffinitefields
chapter4.thestructureoftheresidueclassringmodmandofthereducedresidueclassgroupmodm
1.generalfactsconcerningdirectproductsanddirectsums
2.directdecompositionoftheresidueclassringmodmandofthereducedresidueclassgroupmodm
3.thestructureoftheadditivegroupoftheresidueclassringmodm
4.onthestructureoftheresidueclassringmodpμ
5.thestructureofthereducedresidueclassgroupmodpμ
functionfields
chapter5.quadraticresidues
1.theoryofthecharactersofafiniteabeliangroup
2.residueclasscharactersandnumericalcharactersmodm
3.thebasicfactsconcerningquadraticresidues
4.thequadraticreciprocitylawforthelegendresymbol
5.thequadraticreciprocitylawforthejacobisymbol
6.thequadraticreciprocitylawasproductformulaforthehilbcrtsymbol
7.specialcasesofdirichlet'stheoremonprimenumbersinreducedresidueclasses
functionfield

partⅱ.thetheoryofvaluedfields
chapter6.thefundamentalconceptsregardingvaluations
1.thedefinitionofavaluation；equivalentvaluations
2.approximationindependenceandmultiplicativeindependenceofvaluations
3.valuationsoftheprimefield
4.valuegroupsandresidueclassfields
functionfields
chapter7.arithmeticinadiscretevaluedfield
divisorsfromanideal-theoreticstandpoint
chapter8.thecompletionofavaluedfield
chapter9.thecompletionofadiscretevaluedfield.thelo-adienumberfields
functionfields
chapter10.theisomorphismtypesofcompletediscretevaluedfieldswithperfectresidueclassfield
1.themultiplicativeresiduesysteminthecaseofprimecharacteristic
2.theequal-characteristiccasewithprimecharacteristic
3.themultiplicativeresiduesysteminthep-adicnumberfield
4.witt'svectorcalculus
5.constructionofthegeneralp-adicfield
6.theunequal-characteristiccase
7.isomorphicresiduesystemsinthecaseofcharacteristic0
8.theisomorphicresiduesystemsforarationalfunctionfield
9.theequal-characteristiccasewithcharacteristic0
chapter11.prolongationofadiscretevaluationtoapurelytranscendentalextension
chapter12.prolongationofthevaluationofacompletefieldtoafinitealgebraicextension
1.theproofofexistence
2.theproofofcompleteness
3.theproofofuniqueness
chapter13.theisomorphismtypesofcompletearchimedeanvaluedfields
chapter14.thestructureofafinite-algebraicextensionofacompletediscretevaluedfield
1.embeddingofthearithmetic
2.thetotallyramifiedcase
3.theunramifiedcasewithperfectresidueclassfield
4.thegeneralcasewithperfectresidueclassfield
5.thegeneralcasewithfiniteresidueclassfield
chapter15.thestructureofthemultiplicativegroupofacompletediscretevaluedfieldwithperfectresidueclassfieldofprimecharacteristic
1.reductiontotheone-unitgroupanditsfundamentalchainofsubgroups
2.theone-unitgroupasanabelianoperatorgroup
3.thefieldofnthrootsofunityoverap-adicnumberfield
4.thestructureoftheone-unitgroupintheequal-charaeteristiecasewithfiniteresidueclassfield
5.thestructureoftheone-unitgroupinthep-adiecase
6.constructionofasystemoffundamentalone-unitsinthep-adiccase
7.theone-unitgroupforspecialp-adicnumberfields
8.comparisonofthebasisrepresentationofthemultiplieativegroupinthep-adiccaseandthearchimedeancase
chapter16.thetamelyramifiedextensiontypesofacompletediscretevaluedfieldwithfiniteresidueclassfieldofcharacteristicp
chapter17.theexponentialfunction，thelogarithm，andpowersinacompletenon-archimedeanvaluedfieldofcharacteristic0
1.integralpowerseriesinoneindeterminateoveranarbitraryfield
2.integralpowerseriesinonevariableinacompletenon-archimedeanvaluedfield
3.convergence
4.functionalequationsandmutualrelations
5.thediscretecase
6.theequal-characteristiccasewithcharacteristic0
chapter18.prolongationofthevaluationofanon-completefieldtoafinite-algebraicextension
1.representationsofaseparablefinite-algebraicextensionoveranarbitraryextensionofthegroundfield
2.theringextensionofaseparablefinite-algebraicextensionbyanarbitrarygroundfieldextension，orthetensorproduct
ofthetwofieldextensions
3.thecharacteristicpolynomial
4.supplementsforinseparableextensions
5.prolongationofavaluation
6.thediscretecase
7.thearchimedeancase

partⅲ.thefoundationsofarithmeticinalgebraicnumberfields
chapter19.relations]3etweenthecompletesystemofvaluationsandthearithmeticoftherationalnumberfield
1.finitenessproperties
2.characterizationsindivisibilitytheory
3.theproductformulaforvaluations
4.thesumformulafortheprincipalpartsfunctionfields
theautomorphismsofarationalfunctionfield
chapter20.prolongationofthecompletesystemofvaluationstoafinitealgebraicextension
functionfields
concludingremarks
chapter21.theprimespotsofanalgebraicnumberfieldandtheircompletions
functionfields
chapter22.decompositionintoprimedivisors，integrality，anddivisibility
1.thecanonicalhomomorphismofthemultiplicativegroupintothedivisorgroup
2.embeddingofdivisibilitytheoryunderafinite-algebraicextension
3.algebraiccharacterizationofintegralalgebraicnumbers
4.quotientrepresentation
functionfields
constantfields，constantextensions
chapter23.congruences
1.ordinarycongruence
2.multiplicativecongruence
functionfields
chapter24.themultiplesofadivisor
1.fieldbases
2.theidealproperty，idealbases
3.congruencesforintegralelements
4.divisorsfromtheideal-theoreticstandpoint
5.furtherremarksconcerningdivisorsandideals
functionfields
constantfieldsforp.characterizationofprimedivisorsbyhomomorphisms.decompositionlawunderanalgebraicconstantextension
therankofthemoduleofmultiplesofadivisor
chapter25.differentsanddiscriminants
1.compositionformulaforthetraceandnorm.thedivisortrace
2.definitionofthedifferentanddiseriminant
3.theoremsondifferentsanddiscriminantsinthesmall
4.therelationshipbetweendifferentsanddiscriminantsinthesmallandinthelarge
5.theoremsondifferentsanddiscriminantsinthelarge
6.commoninessentialdiscriminantdivisors
7.examples
functionfields
thenumberoffirst-degreeprimedivisorsinthecaseofafiniteconstantfield
differentials
theriemann-rochtheoremanditsconsequences
disclosedalgebraicfunctionfields
chapter26.quadraticnumberfields
1.generationinthelargeandinthesmall
2.thedecompositionlaw
3.discriminants，integralbases
4.quadraticresiduecharactersofthediscriminantofanarbitraryalgebraicnumberfield
5.thequadraticnumberfieldsasclassfields
6.thehilbertsymbolasnormsymbol
7.thenormtheorem
8.anecessaryconditionforprincipaldivisors.genera
chapter27.cyelotomicfields
1.generation
2.thedecompositionlaw
3.discriminants，integralbases
4.thequadraticnumberfieldsassubfieldsofcyclotomicfields
chapter28.units
1.preliminaries
2.proofs
3.extension
4.examplesandapplications
chapter29.theclassnumber
1.finitenessoftheclassnumber
2.consequences
3.examplesandapplications
functionfields
chapter30.approximationtheoremsandestimatesofthediseriminant
1.themostgeneralrequirementsonapproximatingzero
2.minkowski'slattice-pointtheorem
3.applicationtoconvexbodicewithinthenorm-onehypersurface
4.consequencesofthediscriminantestimate
functionfields
indexofnames
subjectindex