Abstract Algebra An Introduction with Applications 2nd edition by Derek Robinson ISBN 3110340860 9783110340860

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 groups

5.4 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 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

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 radicals

13 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