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Abstract Algebra: An Introduction with Applications (2nd Edition) by Derek J.S. Robinson

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Abstract Algebra: An Introduction with Applications (2nd Edition) written by Prof. Dr. Derek J. S. Robinson , University of Illinois, Department of Mathematics, 1409 West Green Street, Urbana IL 61801, USA, dsrobins@illinois.edu. 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 Algebra: An Introduction with Applications (2nd Edition) written by Derek J.S. Robinson cover the following topics.

• 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 el0
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 2

• 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 nite solvable groups

• 11. The Theory of eld
11.1 Field extensions
11.2 Constructions with ruler and compass
11.3 Finite eld
11.4 Latin squares and Steiner triple systems

• 12. Galois Theory
12.1 Normal and separable extensions
12.2 Automorphisms of eld extension
12.3 The Fundamental Theorem of Galois theory
12.4 Solvability of equations by radicals

• 13. Tensor products
13.1 Denition of the tensor produc
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

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