An Introduction to Abstract Algebra

Description

Derek J. S. Robinson is Professor of Mathematics at the University of Illinois in Urbana-Champaign, USA. 1 Sets, relations and functions 1.1 Sets and subsets 1.2 Relations, equivalence relations and 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 of a set 3.2 Binary operations: semigroups, monoids and groups 3.3 Groups and subgroups 4 Cosets, quotient groups and homomorphisms 4.1 Cosets and Lagrange’s Theorem 4.2 Normal subgroups and quotient groups 4.3 Homomorphisms of groups 5 Groups acting on sets 5.1 Group actions and permutation representations 5.2 Orbits and stabilizers 5.3 Applications to the structure of groups 5.4 Applications to combinatorics – counting labellings and graphs 6 Introduction to rings 6.1 Definition and elementary properties of rings 6.2 Subrings and ideals 6.3 Integral domains, division rings and fields 7 Division in 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 Orthogonality in vector spaces 9 The structure of groups 9.1 The Jordan-Holder Theorem 9.2 Solvable and nilpotent groups 9.3 Theorems on finite solvable groups 10 Introduction to the theory of fields 10.1 Field extensions 10.2 Constructions with ruler and compass 10.3 Finite fields 10.4 Applications to latin squares and Steiner triple systems 11 Galois theory 11.1 Normal and separable extensions 11.2 Automorphisms of field extensions 11.3 The Fundamental Theorem of Galois Theory 11.4 Solvability of equations by radicals 12 Further topics 12.1 Zorn’s Lemma and its applications 12.2 More on roots of polynomials 12.3 Generators and relations for groups 12.4 An introduction to error correcting codes Bibliography Index of notation Index