近世代数概论

PrefacetotheFourthEdition

1TheIntegers

1.1CommutativeRings;IntegralDomains

1.2ElementaryPropertiesofCommutativeRings

1.3OrderedDomains

1.4Well-OrderingPrinciple

1.5FiniteInduction;LawsofExponents

1.6ivisibility

1.7TheEuclideanAlgorithm

1.8FundamentalTheoremofArithmetic

1.9Congruences

1.10TheRingsZn

1.11Sets,Functions,andRelations

1.12IsomorphismsandAutomorphisms

2RationalNumbersandFields

2.1DefinitionofaField

2.2ConstructionoftheRationals

2.3SimultaneousLinearEquations

2.4OrderedFields

2.5PostulatesforthePositiveIntegers

2.6PeanoPostulates

3Polynomials

3.1PolynomialForms

3.2PolynomialFunctions

3.3HomomorphismsofCommutativeRings

3.4PolynomialsinSeveralVariables

3.5TheDivisionAlgorithm

3.6UnitsandAssociates

3.7IrreduciblePolynomials

3.8UniqueFactorizationTheorem

3.9OtherDomainswithUniqueFactorization

3.10EisensteinsIrreducibilityCriterion

3.11PartialFractions

4RealNumbers

4.1DilemmaofPythagoras

4.2UpperandLowerBounds

4.3PostulatesforRealNumbers

4.4RootsofPolynomialEquations

4.5DedekindCuts

5ComplexNumbers

5.1Definition

5.2TheComplexPlane

5.3FundamentalTheoremofAlgebra

5.4ConjugateNumbersandRealPolynomials

5.5QuadraticandCubicEquations

5.6SolutionofQuarticbyRadicals

5.7EquationsofStableType

6Groups

6.1SymmetriesoftheSquare

6.2GroupsofTransformations

6.3FurtherExamples

6.4AbstractGroups

6.5Isomorphism

6.6CyclicGroups

6.7Subgroups

6.8LagrangesTheorem

6.9PermutationGroups

6.10EvenandOddPermutations

6.11Homomorphisms

6.12Automorphisms;ConjugateElements

6.13QuotientGroups

6.14EquivalenceandCongruenceRelations

7VectorsandVectorSpaces

7.1VectorsinaPlane

7.2Generalizations

7.3VectorSpacesandSubspaces

7.4LinearIndependenceandDimension

7.5MatricesandRow-equivalence

7.6TestsforLinearDependence

7.7VectorEquations;HomogeneousEquations

7.8BasesandCoordinateSystems

7.9InnerProducts

7.10EuclideanVectorSpaces

7.11NormalOrthogonalBases

7.12Quotient-spaces

7.13LinearFunctionsandDualSpaces

8TheAlgebraofMatrices

8.1LinearTransformationsandMatrices

8.2MatrixAddition

8.3MatrixMultiplication

8.4Diagonal,Permutation,andTriangularMatrices

8.5RectangularMatrices

8.6Inverses

8.7RankandNullity

8.8ElementaryMatrices

8.9EquivalenceandCanonicalForm

8.10BilinearFunctionsandTensorProducts

8.11Quaternions

9LinearGroups

9.1ChangeofBasis

9.2SimilarMatricesandEigenvectors

9.3TheFullLinearandAffineGroups

9.4TheOrthogonalandEuclideanGroups

9.5InvariantsandCanonicalForms

9.6LinearandBilinearForms

9.7QuadraticForms

9.8QuadraticFormsUndertheFullLinearGroup

9.9RealQuadraticFormsUndertheFullLinearGroup

9.10QuadraticFormsUndertheOrthogonalGroup

9.11QuadricsUndertheAffineandEuclideanGroups

9.12UnitaryandHermitianMatrices

9.13AffineGeometry

9.14ProjectiveGeometry

10DeterminantsandCanonicalForms

10.1DefinitionandElementaryPropertiesofDeterminants

10.2ProductsofDeterminants

10.3DeterminantsasVolumes

10.4TheCharacteristicPolynomial

10.5TheMinimalPolynomial

10.6Cayley-HamiltonTheorem

10.7InvariantSubspacesandReducibility

10.8FirstDecompositionTheorem

10.9SecondDecompositionTheorem

10.10RationalandJordanCanonicalForms

11BooleanAlgebrasandLattices

11.1BasicDefinition

11.2Laws:AnalogywithArithmetic

11.3BooleanAlgebra

11.4DeductionofOtherBasicLaws

11.5CanonicalFormsofBooleanPolynomials

11.6PartialOrderings

11.7Lattices

11.8RepresentationbySets

12TransfiniteArithmetic

12.1NumbersandSets

12.2CountableSets

12.3OtherCardinalNumbers

12.4AdditionandMultiplicationofCardinals

12.5Exponentiation

13RingsandIdeals

13.1Rings

13.2Homomorphisms

13.3Quotient-rings

13.4AlgebraofIdeals

13.5PolynomialIdeals

13.6IdealsinLinearAlgebras

13.7TheCharacteristicofaRing

13.8CharacteristicsofFields

14AlgebraicNumberFields

14.1AlgebraicandTranscendentalExtensions

14.2ElementsAlgebraicoveraField

14.3AdjunctionofRoots

14.4DegreesandFiniteExtensions

14.5IteratedAlgebraicExtensions

14.6AlgebraicNumbers

14.7GaussianIntegers

14.8AlgebraicIntegers

14.9SumsandProductsofIntegers

14.10FactorizationofQuadraticIntegers

15GaloisTheory

15.1RootFieldsforEquations

15.2UniquenessTheorem

15.3FiniteFields

15.4TheGaloisGroup

15.5SeparableandInseparablePolynomials

15.6PropertiesoftheGaloisGroup

15.7SubgroupsandSubfields

15.8IrreducibleCubicEquations

15.9InsolvabilityofQuinticEquations

Bibliography

ListofSpecialSymbols

Index

　　近世代数也称抽象代数，是现代数学的重要基础，主要研究群、环、域等代数结构。本书出自抽象代数领域的两位巨匠之手，曾对近世代数教学产生深远的影响，帮助了几代学子理解和掌握近世代数，至今本书仍是一部对自学和课堂教学都极具价值的参考书和教材。作者用大家热悉且具体的例子来阐述每一个概念，深入浅出，透彻简洁。为了培养学生独立思考的能力，每个专题都包括丰富的练习。　　本书出自近世代数领域的两位科学巨匠之手，是一本经典的教材。全书共分为15章，内容包括：整数、多项式、实数、复数、矩阵代数、线性群、行列式和标准型、布尔代数和格、超限算术、环和理想、代数数域和伽罗华理论等。