Abstract Algebra : an Introduction with Applications.

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Preface; 1 Sets, relations and functions; 1.1 Sets and subsets; 1.2 Relations, equivalence relations, partial orders; 1.3 Functions; 1.4 Cardinality; 2 The integers; 2.1 Well-ordering and mathematical induction; 2.2 Division in the integers; 2.3 Congruences; 3 Introduction to groups; 3.1 Permutations; 3.2 Semigroups, monoids and groups; 3.3 Groups and subgroups; 4 Quotient groups and homomorphisms; 4.1 Cosets and Lagrange's Theorem; 4.2 Normal subgroups and quotient groups; 4.3 Homomorphisms; 5 Groups acting on sets; 5.1 Group actions; 5.2 Orbits and stabilizers.

5.3 Applications to the structure of groups5.4 Applications to combinatorics; 6 Introduction to rings; 6.1 Elementary properties of rings; 6.2 Subrings and ideals; 6.3 Integral domains, division rings and fields; 6.4 Finiteness conditions on ideals; 7 Division in commutative rings; 7.1 Euclidean domains; 7.2 Principal ideal domains; 7.3 Unique factorization in integral domains; 7.4 Roots of polynomials and splitting fields; 8 Vector spaces; 8.1 Vector spaces and subspaces; 8.2 Linear independence, basis and dimension; 8.3 Linear mappings; 8.4 Eigenvalues and eigenvectors.

9 Introduction to modules9.1 Elements of module theory; 9.2 Modules over principal ideal domains; 9.3 Applications to linear operators; 10 The Structure of groups; 10.1 The Jordan-Hölder Theorem; 10.2 Solvable and nilpotent groups; 10.3 Theorems on finite solvable groups; 11 The Theory of fields; 11.1 Field extensions; 11.2 Constructions with ruler and compass; 11.3 Finite fields; 11.4 Latin squares and Steiner triple systems; 12 Galois Theory; 12.1 Normal and separable extensions; 12.2 Automorphisms of field extensions; 12.3 The Fundamental Theorem of Galois theory.

12.4 Solvability of equations by radicals13 Tensor products; 13.1 Definition of the tensor product; 13.2 Properties of tensor products.; 13.3 Extending the ring of operators.; 14 Further topics; 14.1 Zorn's Lemma with applications; 14.2 Roots of polynomials and discriminants; 14.3 Presentations of groups; 14.4 Introduction to error correcting codes; Bibliography; List of symbols; Index.

This is the second edition of the introduction to abstract algebra. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. There is ample material here for a two semester course in abstract algebra.

Includes bibliographical references and index.

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