代数数论讲义
作者：Hecke
出版时间： 2000年版
内容简介
Hecke was certainly one of the masters， and in fact， the study of Hecke Lseries and Hecke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book，and Hecke＇s Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master， Andre Weil： "To improve upon Hecke， in a treatment along classical lines of the theory of algebraic numbers， would be a futile and impossible task."此书为英文版！
目录
CHAPTERI
ElementsofRationalNumberTheory
1.Divisibility,GreatestCommonDivisors,Modules,Prime
Numbers,andtheFundamentalTheoremofNumberTheory
(Theorems1-5)
2.CongruencesandResidueClasses(Euler'sfunction(n).
Ferrnat'stheorem.Theorems6-9)
3.IntegralPolynomials,FunctionalCongruences,andDivisibility
modp(TheoremslO-13a)
4.CongruencesoftheFirstDegree(Theorems14-15)
CHAPTERII
AbelianGroups
5.TheGeneralGroupConceptandCalculationwithElements
ofaGroup(Theorems16-18)
6.SubgroupsandDivisionofaGroupbyaSubgroup(Order
ofelements.Theorems19-21)
7.AbelianGroupsandtheProductofTwoAbeliunGroups
(Theorems22-25)
8.BasisofanAbelianGroup(Thebasisnumberoragroup
belongingtoaprimenumber.Cyclicgroups.Theorems26-28)
9.CompositionofCosetsandtheFactorGroup(Theorem29)
10.CharactersofAbelianGroups(Thegroupofcharacters.
Determinationofallsubgroups.Theorems30-33)
11.InfiniteAbelianGroups(Finitebasisofsuchagroupand
basisforasubgroup.Theorems34-40)
CHAPTERIII
AbelianGroupsinRationalNumberTheory
12.GroupsofIntegersunderAdditionandMultiplication
(Theorem41)
13.StructureoftheGroupR(n)oftheResidueClassesmodn
RelativelyPrimeton(Primitivenumbersmodpandmodp2.
Theorems42-45)
14.PowerResidues(Binomialcongruences.Theorems46-47)
15.ResidueCharactersofNumbersmodn
16.QuadraticResidueCharactersmodn(Onthequadratic
reciprocitylaw)
CHAPTERIV
AlgebraofNumberFields
17.NumberFields,PolynomialsoverNumberFields,and
Irreducibility(Theorems48-49)
18.AlgebraicNumbersoverk(Theorems50-519
19.AlgebraicNumberFieldsoverk(Simultaneousad)unctionof
severalnumbers.Theconjugatenumbers.Theorems52-55)
20.GeneratingFieldElements,FundamentalSystems,and
SubfieldsofK(0)(Theorems56-59)
CHAPTERV
GeneralArithmeticofAlgebraicNumberFields
21.DefinitionofAlgebraicIntegers,Divisibility,andUnits
(Theorems60-63)
22.TheIntegersofaFieldasanAbelianGroup:Basisand
DiscriminantoftheField(Moduli.Theorem64)
23.FactorizationofIntegersinK():GreatestCommon
DivisorswhichDoNotBelongtotheField
24.DefinitionandBasicPropertiesofIdeals(Productofideals.
Primeideals.Twodefinitionsofdivisibility.Theorems65-69)
25.TheFundamentalTheoremofIdealTheory(Theorems70-72)
26.FirstApplicationsoftheFundamentalTheorem(Theorems73-75)
27.CongruencesandResidueClassesModuloIdealsandthe
GroupofResidueClassesunderAdditionandunder
Multiplication(Normofanideal.Fermat'stheoremforideal
theory.Theorems76-85)
28.PolynomialswithIntegralAlgebraicCoefficients(Contentof
polynomials.Theorems86-87)
29.FirstTypeofDecompositionLawsforRationalPrimes:
DecompositioninQuadraticFields(Theorems88-90)
30.SecondTypeofDecompositionTheoremforRationalPrimes:
DecompositionintheFieldK(e2xi/m)(Theorems91-92)
31.FractionalIdeals(Theorem93)
32.Minkowski'sTheoremonLinearForms(Theorems94-95)
33.IdealClasses,theClassGroup,andIdealNumbers
(Theorems96-98)
34.UnitsandanUpperBoundfortheNumberofFundamental
Units(Theorems99-100)
35.Dirichlet'sTheoremabouttheExactNumberofFundamental
Units(Theregulatorofthefield)
36.DifferentandDiscriminant(Numberrings.Theorems
101-105)
37.RelativeFieldsandRelationsbetweenIdealsinDifferentFields
(Theorem106J
38.RelativeNorms'ofNumbersandIdeals,RelativeDifferents,and
RelativeDiscriminants(Theprimefactorsoftherelative
different.Theorems107-115)
39.DecompositionLawsintheRelativeFieldsK()(Theorems
116-120)
CHAPTERVI
IntroductionofTranscendentalMethodsintothe
ArithmeticofNumberFields
40.TheDensityoftheIdealsinaClass(Theorem121)
41.TheDensityofIdealsandtheClassNumber(Thenumber
ofidealswithgivennorm.Theorem122)
42.TheDedekindZeta-Function(Dirichletseries.Dedekind's
zeta-functionanditsbehaviorats=1.Representationby
products.Theorems123-125)
43.TheDistributionofPrimeIdealsofDegree1,inParticularthe
RationalPrimesinArithmeticProgressions(TheDirichlet
serieswithresiduecharactersmodn.Degreeofthecyciotomic
fields.Theorems126-131)
CHAPTERVII
TheQuadraticNumberField
44.SummaryandtheSystemofIdealClasses(Numericalexamples)
45.TheConceptofStrictEquivalenceandtheStructureofthe
ClassGroup(Theorems132-134)
46.TheQuadraticReciprocityLawandaNewFormulationofthe
DecompositionLawsinQuadraticFields(Theorems135-137)
47.NormResiduesandtheGroupofNormsofNumbers
(Theorems138-141)
48.TheGroupofIdealNorms,theGroupofGenera,and
DeterminationoftheNumberofGenera(Theorems142-145)
49.TheZeta-Functionofk()andtheExistenceofPrimeswith
PrescribedQuadraticResidueCharacters(Theorems
146-147)
50.DeterminationoftheClassNumberofk()withoutUsaofthe
Zeta-Function(Theorem148)
54.DeterminationoftheClassNumberwiththeHelpofthe
Zeta-Function(Theorem149)
52.GaussSumsandtheFinalFormulafortheClassNumber
(Theorems150-152)
53.ConnectionbetweenIdealsink()andBinaryQuadratic
Forms(Theorems153-154)
CHAPTERVIII
TheLawofQuadraticReciprocityinArbitrary
NumberFields
54.QuadraticResidueCharactersandGaussSumsinArbitrary
NumberFields(Theorems155-156)
55.Theta-functionsandTheirFourierExpansions(Theorems
157-158)
56.ReciprocitybetweenGaussSumsinTotallyRealFields(The
transformationformulaofthethetafunctionandthereciprocity
betweenGaussstansfortotallyrealfields.Theorems159-161)
57.ReciprocitybetweenGaussSumsinArbitraryAlgebraic
NumberFields(Thetransformationformulaofthetheta
functionandthereciprocitybetweenGausssumsforarbitrary
fields.Theorems162-163)
58.TheDeterminationoftheSignofGaussSumsintheRational
NumberField(Theorem164)
59.TheQuadraticReciprocityLawandtheFirstPartofthe
SupplementaryTheorem(Theorems165-167)
60.RelativeQuadraticFieldsandApplicationstotheTheoryof
QuadraticResidues(Existenceofprimeidealswith
prescribedresiduecharacters.Theorems168-169)
61.NumberGroups,IdealGroups,andSingularPrimaryNumbers
61.NumberGroups,IdealGroups,andSingularPrimaryNumbers
62.TheExistenceoftheSingularPrimaryNumbersand
SupplementaryTheoremsfortheReciprocityLaw(Theorems
170-175)
63.APropertyofFieldDifferentsandtheHilbertClassFieldof
RelativeDegree2(Theorems176-179)
ChronologicalTable
References
