Fundamentals of Abstract Algebra

CHAPTER 1: Sets, Relations, and IntegersSetsIntegersRelationsPartially Ordered SetFunctionsBinary OperationsCHAPTER 2: Introduction to GroupsElementary Properties of GroupsCHAPTER 3: Permution GroupsPermution GroupsCHAPTER 4: Subgroups and Normal SubgroupsSubgroupsCyclic GroupsLagrange's TheoremNormal Subgroups and Quotient GroupsCHAPTER 5: Homomorphisms and Isomorphisms of GroupsHomomorphisms of GroupsIsomorphisms and Correspondence TheoremsGroups D4 and Q8Group ActionsCHAPTER 7: Sylow TheoremConjugacy ClassesCauchy Theorem and p-groupsSylow TheoremsSome Applications of the Sylow TheoremCHAPTER 8: Solvable and Nilpotent GroupsSolvable Groups Nilpotent GroupsCHAPTER 9: Finitely Generated Abelian GroupsFinite Abelian GroupsFinitely Generated Abelian GroupsCHAPTER 10: Introduction to RingsElementary PropertiesSome Important RingsCHAPTER 11: Subrings, Ideals, and HomomorphismsSubrings and SubfieldsIdeals and Quotient RingsHomomorphisms and IsomrophismsCHAPTER 12: Extensions of RingsExtensions of RingsCHAPTER 13: Direct Sum of RingsComplete Direct Sum and Direct SumCHAPTER 14: Polynomial RingsPolynomial RingsCHAPTER 15: Euclidean DomainsEuclidean DomainsGreatest Common DivisorsPrime and Irreducible ElementsCHAPTER 16: Unique Factorization DomainsUnique Factorization DomainsFactorization of Polynomials over a UFDIrreducibility of PolynomialsCHAPTER 17: Maximal, Prime, and Primary IdealsMaximal, Prime and Primary IdealsJacobson Semisimple RingCHAPTER 18: Noetherian and Artinian RingsNoetherian and Artinian RingsCHAPTER 19: Modules and Vector SpacesModules and Vector SpacesCHAPTER 20: Matrix RingsFull Matrix Rings Triangular Matrix RingsCHAPTER 21: Field ExtensionsAlgebraic ExtensionsSplitting FieldsAlgebraically Colsed FieldsCHAPTER 22: Multiplicity of RootsMultiplicity of RootsCHAPTER 23: Finite FieldsFinite FieldsCHAPTER 24: Galois Theory and ApplicationsNormal ExtensionsGalois TheoryRoots of Unity and Cyclotomic PolynomialsSolution by RadicalsCHAPTER 25: Geometric Constructions Feometric ConstructionsCHAPTER 26: Binary CodesBinary CodesPolynomial and Cyclic CodesBose-Chauduri-Hocquenghem CodesCHAPTER 27: Groebner BasesAffine VaritiesGroebner BasesSelected BibliographyAnswers and Hints to Selected ExercisesIndex