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ABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONS 3 ABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONS 5 Library of Congress Cataloging-in-Publication Data 6 Contents 7 Introduction 9 Chapter 1: Tools and Major Results of Groups 11 1.1 Notations 11 1.2 Results 12 Chapter 2: Problems in Group Theory 17 2.1 Elementary Properties of Groups 17 2.2 Subgroups 21 2.3 Cyclic Groups 24 2.4 Permutation Groups 28 2.5 Cosets and Lagrange's Theorem 31 2.6 Normal Subgroups and Factor Groups 36 2.7 Group Homomorphisms and Direct Product 44 2.8 Sylow Theorems 57 Chapter 3: Tools and Major Results of Ring Theory 65 3.1 Notations 65 3.2 Major Results of Ring Theory 66 Chapter 4: Problems in Ring Theory 71 4.1 Basic Properties of Rings 71 4.2 Ideals, Subrings, and Factor Rings 75 4.3 Integral Domains, and Zero Divisors 83 4.4 Ring Homomorphisms and Ideals 87 4.5 Polynomial Rings 94 4.6 Factorization in Polynomial Rings 100 4.7 Extension Fields, and Algebraic Fields 102 4.8 Finite Fields 108 Bibliography 115 Index 117

Author

Owner

Published

2013

Language

EN