Sylow Theorems with proofs and applications. Non-simplicity tests and techniques for showing that certain numbers cannot be the orders of simple groups. Introduction to group rings. Euclidean domains illustrated by the integers, Gaussian integers and polynomials in one indeterminate over a field. Finding highest common factors in a Euclidean domain.
Every Euclidean domain is a principal ideal domain.
Maximal and prime ideals. Irreducibles and primes, factorisation in principal ideal domains and unique factorisation domains.