Instructor’sSolutions Manualto accompanyA First Course inAbstract AlgebraSeventh EditionJohn B. FraleighUniversity of Rhode IslandFull file at

Full file at

PrefaceThis manual contains solutions to all exercises in the text, except those odd-numbered exercises for whichfairly lengthy complete solutions are given in the answers at the back of the text. Then reference is simplygiven to the text answers to save typing.I prepared these solutions myself.While I tried to be accurate, there are sure to be the inevitablemistakes and typos.An author reading proof rends to see what he or she wants to see.However, theinstructor should find this manual adequate for the purpose for which it is intended.Morgan, VermontJ.B.FJuly, 2002iFull file at

iiFull file at

CONTENTS0. Sets and Relations1I. Groups and Subgroups1. Introduction and Examples42. Binary Operations73. Isomorphic Binary Structures94. Groups135. Subgroups176. Cyclic Groups217. Generators and Cayley Digraphs24II. Permutations, Cosets, and Direct Products8. Groups of Permutations269. Orbits, Cycles, and the Alternating Groups3010. Cosets and the Theorem of Lagrange3411. Direct Products and Finitely Generated Abelian Groups3712. Plane Isometries42III. Homomorphisms and Factor Groups13. Homomorphisms4414. Factor Groups4915. Factor-Group Computations and Simple Groups5316. Group Action on a Set5817. Applications of G-Sets to Counting61IV. Rings and Fields18. Rings and Fields6319. Integral Domains6820. Fermat’s and Euler’s Theorems7221. The Field of Quotients of an Integral Domain7422. Rings of Polynomials7623. Factorization of Polynomials over a Field7924. Noncommutative Examples8525. Ordered Rings and Fields87V. Ideals and Factor Rings26. Homomorphisms and Factor Rings8927. Prime and Maximal Ideals9428. Gr¨obner Bases for Ideals99iiiFull file at

VI. Extension Fields29. Introduction to Extension Fields10330. Vector Spaces10731. Algebraic Extensions11132. Geometric Constructions11533. Finite Fields116VII. Advanced Group Theory34. Isomorphism Theorems11735. Series of Groups11936. Sylow Theorems12237. Applications of the Sylow Theory12438. Free Abelian Groups12839. Free Groups13040. Group Presentations133VIII. Groups in Topology41. Simplicial Complexes and Homology Groups13642. Computations of Homology Groups13843. More Homology Computations and Applications14044. Homological Algebra144IX. Factorization45. Unique Factorization Domains14846. Euclidean Domains15147. Gaussian Integers and Multiplicative Norms154X. Automorphisms and Galois Theory48. Automorphisms of Fields15949. The Isomorphism Extension Theorem164

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