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Abstract Algebra An Introduction with Applications (2nd Edition) by Derek John Scott Robinson

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Abstract Algebra An Introduction with Applications (2nd Edition) written by Derek John Scott Robinson . 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 sample material here for a two semester course in abstract algebra

Abstract Algebra An Introduction with Applications (2nd Edition) written by Derek John Scott 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 eld
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 eld

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 groupsÊ

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

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

13. Tensor products
13.1 Denition 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

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