Solutions Manual to accompany Introduction to Abstract Algebra, 4e

W. KEITH NICHOLSON, PhD, is Professor in the Department of Mathematics and Statistics at the University of Calgary, Canada. He has published extensively in his areas of research interest, which include clean rings, morphic rings and modules, and quasi-morphic rings.

• 0 Preliminaries 10.1 Proofs 10.2 Sets 20.3 Mappings 30.4 Equivalences 41 Integers and Permutations 61.1 Induction 61.2 Divisors and Prime Factorization 81.3 Integers Modulo 111.4 Permutations 132 Groups 172.1 Binary Operations 172.2 Groups 192.3 Subgroups 212.4 Cyclic Groups and the Order of an Element 242.5 Homomorphisms and Isomorphisms 282.6 Cosets and Lagrange's Theorem 302.7 Groups of Motions and Symmetries 322.8 Normal Subgroups 342.9 Factor Groups 362.10 The Isomorphism Theorem 382.11 An Application to Binary Linear Codes 433 Rings 473.1 Examples and Basic Properties 473.2 Integral Domains and Fields 523.3 Ideals and Factor Rings 553.4 Homomorphisms 593.5 Ordered Integral Domains 624 Polynomials 644.1 Polynomials 644.2 Factorization of Polynomials over a Field 674.3 Factor Rings of Polynomials over a Field 704.4 Partial Fractions 764.5 Symmetric Polynomials 765 Factorization in Integral Domains 815.1 Irreducibles and Unique Factorization 815.2 Principal Ideal Domains 846 Fields 886.1 Vector Spaces 886.2 Algebraic Extensions 906.3 Splitting Fields 946.4 Finite Fields 966.5 Geometric Constructions 986.7 An Application to Cyclic and BCH Codes 997 Modules over Principal Ideal Domains 1027.1 Modules 1027.2 Modules over a Principal Ideal Domain 1058 p-Groups and the Sylow Theorems 1088.1 Products and Factors 1088.2 Cauchy’s Theorem 1118.3 Group Actions 1148.4 The Sylow Theorems 1168.5 Semidirect Products 1188.6 An Application to Combinatorics 1199 Series of Subgroups 1229.1 The Jordan-H¨older Theorem 1229.2 Solvable Groups 1249.3 Nilpotent Groups 12710 Galois Theory 13010.1 Galois Groups and Separability 13010.2 The Main Theorem of Galois Theory 13410.3 Insolvability of Polynomials 13810.4 Cyclotomic Polynomials and Wedderburn's Theorem 14011 Finiteness Conditions for Rings and Modules 14211.1 Wedderburn's Theorem 14211.2 The Wedderburn-Artin Theorem 143Appendices 147Appendix A: Complex Numbers 147Appendix B: Matrix Arithmetic 148Appendix C: Zorn's Lemma 149

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