MATH20310 Groups, Rings and Fields

Academic Year 2021/2022

This course will be an introduction to group theory, ring theory and field theory. We will cover the following topics: definition and examples of groups, subgroups, cosets and Lagrange's Theorem, the order of an element of a group, normal subgroups and quotient groups, group homomorphisms and the homomorphism theorem, more isomorphism theorems, definitions of a commutative ring with unity, integral domains and fields, units, irreducibles and primes in a ring, ideals and quotient rings, prime and maximal ideals, ring homorphisms and the homomorphism theorem, polynomial rings, the division algorithm, gcd for polynomials, irreducible polynomials and field extensions. Time permitting, we may cover the Sylow theorems, solvable groups and further examples of groups.