A First Course in Abstract Algebra

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Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. FeaturesThis classical approach to abstract algebra focuses on applications. The text is geared toward high-level courses at schools with strong mathematics programs. Accessible pedagogy includes historical notes written by Victor Katz, an authority on the history of math. By opening with a study of group theory, this text provides students with an easy transition to axiomatic mathematics.Table Of Contents GROUPS AND SUBGROUPS. Introduction and Examples. Binary Operations. Isomorphic Binary Structures. Groups. Subgroups. Cyclic Groups. Generators and Cayley Digraphs. PERMUTATIONS, COSETS, AND DIRECT PRODUCTS. Groups of Permutations. Orbits, Cycles, and the Alternating Groups. Cosets and the Theorem of Lagrange. Direct Products and Finitely Generated Abelian Groups. Plane Isometries. HOMOMORPHISMS AND FACTOR GROUPS. Homomorphisms. Factor Groups. Factor-Group Computations and Simple Groups. Group Action on a Set. Applications of G-Sets to Counting. RINGS AND FIELDS. Rings and Fields. Integral Domains. Fermat's and Euler's Theorems. The Field of Quotients of an Integral Domain. Rings of Polynomials. Factorization of Polynomials over a Field. Noncommutative Examples. Ordered Rings and Fields. IDEALS AND FACTOR RINGS. Homomorphisms and Factor Rings. Prime and Maximal Ideas. Gröbner Bases for Ideals. EXTENSION FIELDS. Introduction to Extension Fields. Vector Spaces. Algebraic Extensions. Geometric Constructions. Finite Fields. ADVANCED GROUP THEORY. Isomorphism Theorems. Series of Groups. Sylow Theorems. Applications of the Sylow Theory. Free Abelian Groups. Free Groups. Group Presentations. GROUPS IN TOPOLOGY. Simplicial Complexes and Homology Groups. Computations of Homology Groups. More Homology Computations and Applications. Homological Algebra. FACTORIZATION. Unique Factorization Domains. Euclidean Domains. Gaussian Integers and Multiplicative Norms. MORPHISMS AND GALOIS THEORY. Automorphisms of Fields. The Isomorphism Extension Theorem. Splitting Fields. Separable Extensions. Totally Inseparable Extensions. Galois Theory. Illustrations of Galois Theory. Cyclotomic Extensions. Insolvability of the Quintic.