This course develops in the theme of "Arithmetic congruence, and abstract algebraic structures." There will be a very strong emphasis on theory and proofs.
Suggested text book: Abstract Algebra, An Introduction, by T. Hungerford
- Review of some of the preliminary material. Basic set, logic, and proof terminology. Well-ordering principle, and equivalence relations - 4 lectures
- Arithmetic in Z and congruence in Z. Division algorithm, congruence and congruence classes, modular arithmetic, and the structure of Z_p when p is a prime
- Rings, fields, and polynomial ring F[x]. Definitions, examples, and basic properties of rings, integral domains, fields, ideals, congruences, quotient rings, homomorphisms and isomorphisms, fields of quotients. Division algorithm, irreducibles and unique factorization in F[x]. Polynomial functions and congruences in F[x]. The structure of F[x]/(\pi) when \pi is a prime in F[x] - 12 lectures
- Groups. Definitions, examples and basic properties of groups, subgroups, normal subgroups. Isomorphisms and homomorphisms, quotient groups, congruence and Lagrange's theorem, the symmetric and alternating groups - 12 lectures
- Other Topics. Finite abelian groups. The Sylow theorems. Splitting fields and finite fields - 12 lectures