Abstract Algebra: An introduction with Applications

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.