代數(shù)數(shù)論講義（英文本）

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