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A course in algebra / Yun Fan, Q.Y. Xiong, Y.L. Zheng.

By: Fan, YunContributor(s): Xiong, Q. Y | Zheng, Y. LMaterial type: TextTextPublication details: Singapore ; River Edge, N.J. : World Scientific, ©2000. Description: 1 online resource (x, 246 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9789812815095; 9812815090Subject(s): Algebra, Abstract | MATHEMATICS -- Algebra -- Abstract | Algebra, Abstract | Algebra | Einführung | Algebra | Álgebra | Álgebra abstrataGenre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: Course in algebra.DDC classification: 512/.02 LOC classification: QA162 | .F36 2000ebOther classification: 31.20 | SK 200 Online resources: Click here to access online
Contents:
1 Preliminaries 1 -- 1.1 Sets 1 -- 1.2 Logic 4 -- 1.3 Relations 7 -- 1.5 Zorn's Lemma 18 -- 2 Groups 19 -- 2.1 Transformations and Permutations 19 -- 2.2 Groups 27 -- 2.3 Subgroups 35 -- 2.4 Homomorphisms, Isomorphisms 39 -- 2.5 Cosets 45 -- 2.6 Normal Subgroups, Quotient Groups 49 -- 2.7 Homomorphism Theorems 52 -- 2.8 Cyclic Groups, Orders of Elements 56 -- 2.9 Direct Products 62 -- 3 Rings 71 -- 3.1 Fundamentals 71 -- 3.2 Zero Divisors, Invertible Elements 77 -- 3.3 Ideals, Residue Rings 81 -- 3.4 Homomorphism Theorems 86 -- 3.5 Prime Ideals, Maximal Ideals 89 -- 3.6 Direct Sums 93 -- 3.7 Fraction Fields of Integral Domains 97 -- 3.8 Polynomial Rings 101 -- 3.9 Factorial Domains 110 -- 3.10 Polynomial Rings over Factorial Rings 117 -- 4 Modules 123 -- 4.1 Modules 123 -- 4.2 Homomorphisms 127 -- 4.3 Direct Products, Direct Sums 133 -- 4.4 Exact Sequences of Homomorphisms 137 -- 4.5 Free Modules, Matrices over Rings 142 -- 4.6 Matrices over Division Rings 150 -- 4.7 Matrices over Commutative Rings 159 -- 4.8 Algebras over Commutative Rings 164 -- 4.9 Tensor Products 171 -- 4.10 Projective Modules, Injective Modules 176 -- 5 Fields 183 -- 5.1 Subfields and Extensions 183 -- 5.2 Single Extensions 187 -- 5.3 Algebraic Extensions 192 -- 5.4 Splitting Fields, Normal Extensions 196 -- 5.5 Two Applications 202 -- 5.6 Separability, Multiple Roots 207 -- 5.7 Finite Fields 215 -- 5.8 Coding 220 -- 5.9 p-adic Numbers 229 -- 5.10 Quaternions 238.
Summary: This volume is based on the lectures given by the authors at Wuhan University and Hubei University in courses on abstract algebra. It presents the fundamental concepts and basic properties of groups, rings, modules and fields, including the interplay between them and other mathematical branches and applied aspects.
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Includes index.

1 Preliminaries 1 -- 1.1 Sets 1 -- 1.2 Logic 4 -- 1.3 Relations 7 -- 1.5 Zorn's Lemma 18 -- 2 Groups 19 -- 2.1 Transformations and Permutations 19 -- 2.2 Groups 27 -- 2.3 Subgroups 35 -- 2.4 Homomorphisms, Isomorphisms 39 -- 2.5 Cosets 45 -- 2.6 Normal Subgroups, Quotient Groups 49 -- 2.7 Homomorphism Theorems 52 -- 2.8 Cyclic Groups, Orders of Elements 56 -- 2.9 Direct Products 62 -- 3 Rings 71 -- 3.1 Fundamentals 71 -- 3.2 Zero Divisors, Invertible Elements 77 -- 3.3 Ideals, Residue Rings 81 -- 3.4 Homomorphism Theorems 86 -- 3.5 Prime Ideals, Maximal Ideals 89 -- 3.6 Direct Sums 93 -- 3.7 Fraction Fields of Integral Domains 97 -- 3.8 Polynomial Rings 101 -- 3.9 Factorial Domains 110 -- 3.10 Polynomial Rings over Factorial Rings 117 -- 4 Modules 123 -- 4.1 Modules 123 -- 4.2 Homomorphisms 127 -- 4.3 Direct Products, Direct Sums 133 -- 4.4 Exact Sequences of Homomorphisms 137 -- 4.5 Free Modules, Matrices over Rings 142 -- 4.6 Matrices over Division Rings 150 -- 4.7 Matrices over Commutative Rings 159 -- 4.8 Algebras over Commutative Rings 164 -- 4.9 Tensor Products 171 -- 4.10 Projective Modules, Injective Modules 176 -- 5 Fields 183 -- 5.1 Subfields and Extensions 183 -- 5.2 Single Extensions 187 -- 5.3 Algebraic Extensions 192 -- 5.4 Splitting Fields, Normal Extensions 196 -- 5.5 Two Applications 202 -- 5.6 Separability, Multiple Roots 207 -- 5.7 Finite Fields 215 -- 5.8 Coding 220 -- 5.9 p-adic Numbers 229 -- 5.10 Quaternions 238.

Print version record.

This volume is based on the lectures given by the authors at Wuhan University and Hubei University in courses on abstract algebra. It presents the fundamental concepts and basic properties of groups, rings, modules and fields, including the interplay between them and other mathematical branches and applied aspects.

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