Abstract Algebra : an Introduction with Applications

Description

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. Abstract: 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 Content: 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.