Abstract Algebra : an Introduction with Applications

✍ Scribed by Robinson, Derek John Scott

• Publisher: De Gruyter
• Year: 2015
• Tongue: English
• Leaves: 347
• Series: De Gruyter textbook
• Edition: 2ed.
• Category: Library

✦ Synopsis

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

✦ Table of Contents

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.

✦ Subjects

Algebra, Abstract -- Textbooks;MATHEMATICS -- Algebra -- Intermediate;Algebra, Abstract