Contemporary Abstract Algebra

Content: Part 1. Integers and equivalence relations. Preliminaries. Properties of integers --
Modular arithmetic --
Complex numbers --
Mathematical induction --
Equivalence relations --
Functions (mappings) --
Exercises --
Part 2. Groups. 1. Introduction to groups. Symmetries of a square --
The dihedral groups --
Exercises --
Biography of Niels Abel --
2. Groups. Definition and examples of groups --
Elementary properties of groups --
Historical note --
Exercises --
3. Finite groups
 subgroups. Terminology and notation --
Subgroup tests --
Examples of subgroups --
Exercises --
4. Cyclic groups. Properties of cyclic groups --
Classification of subgroups of cyclic groups --
Exercises --
Biography of James Joseph Sylvester --
5. Permutation groups. Definition and notation --
Cycle notation --
Properties of permutations --
A check-digit scheme based on D₅ --
Exercises --
Biography of Augustin Cauchy --
Biography of Alan Turing --
6. Isomorphisms. Motivation --
Definition and examples --
Cayley\'s Theorem --
Properties of isomorphisms --
Automorphisms --
Exercises --
Biography of Arthur Cayley --
7. Cosets and Lagrange\'s Theorem. Properties of Cosets --
Lagrange\'s Theorem and consequences --
An application of Cosets to permutation groups --
The rotation group of a cube and a soccer ball --
An application of Cosets to the Rubik\'s cube --
Exercises --
Biography of Joseph Lagrange --
8. External direct products. Definition and examples --
Properties of external direct products --
The group of units modulo n as an external direct product --
Applications --
Exercises --
Biography of Leonard Adleman --
9. Normal subgroups and factor groups. Normal subgroups --
Factor groups --
Applications of factor groups --
Internal direct products --
Exercises --
Biography of Evariste Galois --
10. Group homomorphisms. Definition and examples --
Properties of homomorphisms --
The first isomorphism theorem --
Exercises --
Biography of Camille Jordan --
11. Fundamental theorem of finite Abelian groups. The fundamental theorem --
The isomorphism classes of Abelian groups --
Proof of the fundamental theorem --
Exercises --
Part 3. Rings. 12. Introduction to rings. Motivation and definition --
Examples of rings --
Properties of rings --
Subrings --
Exercises --
Biography of I.N. Herstein --
13. Integral domains. Definition and examples --
Fields --
Characteristic of a ring --
Exercises --
Biography of Nathan Jacobson --
14. Ideals and factor rings. Ideals --
Factor rings --
Prime ideals and maximal ideals --
Exercises --
Biography of Richard Dedekind --
Biography of Emmy Noether --
15. Ring homomorphisms. Definition and examples --
Properties of ring homomorphisms --
The field of quotients --
Exercises --
Biography of Irving Kaplansky --
16. Polynomial rings. Notation and terminology --
The division algorithm and consequences --
Exercises --
Biography of Saunders Mac Lane --
17. Factorization of polynomials. Reducibility tests --
Irreducibility tests --
Unique factorization in Z[x] --
Weird dice : an application of unique factorization --
Exercises --
Biography of Serge Lang --
18. Divisibility in integral domains. Irreducibles, primes --
Historical discussion of Fermat\'s last theorem --
Unique factorization domains --
Euclidean domains --
Exercises --
Biography of Sophie Germain --
Biography of Andrew Wiles --
Biography of Pierre de Fermat --
Part 4. Fields. 19. Vector spaces. Definition and examples --
Subspaces --
Linear independence --
Exercises --
Biography of Emil Artin --
Biography of Olga Taussky-Todd --
20. Extension fields. The fundamental theorem of field theory --
Splitting fields --
Zeros of an irreducible polynomial --
Exercises --
Biography of Leopold Kronecker --
21. Algebraic extensions. Characterization of extensions --
Finite extensions --
Properties of algebraic extensions --
Exercises --
Biography of Ernst Steinitz --
22. Finite fields. Classification of finite fields --
Structure of finite fields --
Subfields of a finite field --
Exercises --
Biography of L.E. Dickson --
23. Geometric constructions. Historical discussion of geometric constructions --
Constructible numbers --
Angle-trisectors and circle-squarers --
Exercises --
Part 5. Special topics. 24. Sylow theorems. Conjugacy classes --
The class equation --
The Sylow theorems --
Applications of Sylow theorems --
Exercises --
Biography of Oslo Ludwig Sylow --
25. Finite simple groups. Historical background --
Nonsimplicity tests --
The simplicity of A₅ --
The Fields Medal --
The Cole Prize --
Exercises --
Biography of Michael Aschbacher --
Biography of Daniel Gorenstein --
Biography of John Thompson --
26. Generators and relations. Motivation --
Definitions and notation --
Free group --
Generators and relations --
Classification of groups of order up to 15 --
Characterization of dihedral groups --
Realizing the dihedral groups with mirrors --
Exercises --
Biography of Marshall Hall, Jr. --
27. Symmetry groups. Isometries --
Classification of finite plane symmetry groups --
Classification of finite groups of rotations in R³ --
Exercises --
28. Frieze groups and crystallographic groups. The frieze groups --
The crystallographic groups --
Identification of plane periodic patterns --
Exercises --
Biography of M.C. Escher --
Biography of George Pólya --
Biography of John H. Conway --
29. Symmetry and counting. Motivation --
Burnside\'s Theorem --
Applications --
Group action --
Exercises --
Biography of William Burnside --
30. Cayley digraphs of groups. Motivation --
The Cayley digraph of a group --
Hamiltonian circuits and paths --
Some applications --
Exercises --
Biography of William Rowan Hamilton --
Biography of Paul Erdos --
31. Introduction to algebraic coding theory. Motivation --
Linear codes --
Parity-check matrix decoding --
Coset decoding --
Historical note : the ubiquitous Reed-Solomon codes --
Exercises --
Biography of Richard W. Hamming --
Biography of Jessie MacWilliams --
Biography of Vera Pless --
32. An introduction to Galois Theory. Fundamental Theorem of Galois Theory --
Solvability of polynomials by radicals --
Insolvability of a quintic --
Exercises --
Biography of Philip Hall --
33. Cyclotomic extensions. Motivation --
Cyclotomic polynomials --
The constructible regular n-gons --
Exercises --
Biography of Carl Friedrich Gauss --
Biography of Manjul Bhargava.