Abstract Algebra: An Introduction with Applications (De Gruyter Textbook)

✍ Scribed by Derek J.S. Robinson

• Publisher: De Gruyter
• Year: 2022
• Tongue: English
• Leaves: 457
• Edition: 3rd, extended
• Category: Library

✦ Synopsis

This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and the information and physical sciences. In addition to introducing the main concepts of modern algebra – groups, rings, modules and fields – the book contains numerous applications, which are intended to illustrate the concepts and to show the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems, error correcting codes and economics are described. There is ample material here for a two semester course in abstract algebra. Proofs of almost all results are given. The reader led through the proofs in gentle stages. There are more than 500 problems, of varying degrees of diffi culty. The book should be suitable for advanced undergraduate students in their fi nal year of study and for fi rst or second year graduate students at a university in Europe or North America. In this third edition three new chapters have been added: an introduction to the representation theory of fi nite groups, free groups and presentations of groups, an introduction to category theory.

✦ Table of Contents

Preface
Contents
List of symbols
1 Sets, Relations and Functions
1.1 Sets and subsets
1.2 Relations, equivalence relations, partial orders
1.3 Functions
1.4 Cardinality
1.5 Zorn’s Lemma and variants
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 groups
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 Modules
9.1 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
12 Galois Theory
12.1 Normal and separable extensions
12.2 Automorphisms of fields
12.3 The Fundamental Theorem of Galois theory
12.4 Solvability of equations by radicals
12.5 Roots of Polynomials and Discriminants
13 Tensor Products
13.1 Definition of the tensor product
13.2 Properties of tensor products
13.3 Extending the ring of operators
14 Representations of groups
14.1 Representations and group rings
14.2 The structure of group algebras
14.3 Group characters
14.4 The Burnside p − q Theorem
15 Presentations of groups
15.1 Free groups
15.2 Generators and relations
15.3 Free products
16 Introduction to category theory
16.1 Categories
16.2 Functors
16.3 Categorical constructions
16.4 Natural transformations
17 Applications
17.1 Set labelling problems
17.2 Enumerating graphs
17.3 Latin squares and Steiner systems
17.4 Introduction to error correcting codes
17.5 Algebraic models for accounting systems
Bibliography
Index