The module introduces the key abstract algebraic objects of groups, rings and fields and develops their fundamental theory. The theory will be illustrated and made concrete through numerous examples in settings that students will already have encountered.


To introduce basic principles of abstract algebraic structures and to teach how an extensive and important theory can be developed by logical deductions from a small number of axioms.

Learning Outcomes

On completion of the course, students should
* Know and understand the formal definitions for Groups, Rings, and Fields
* be able to produce simple proofs based on the algebraic axioms
* Be familiar with standard examples of these algebras, including the Symmetric Group, Modular Arithmetic, finite abelian groups, Polynomial and Matrix Rings, and examples of finite fields.
* Be familiar with the notions of subalgebras as they apply to Groups, Rings, and Fields
* Understand the notion of isomorphism and homomorphism of these algebra types


Groups: Binary operations, groups, subgroups, cyclic groups, direct products, groups of permutations, cosets, Lagrange's theorem; Isomorphisms and homomorphisms of groups

Rings, Fields, zero divisors and integral domains, subrings, ideals. Direct products, homomorphisms, Isomorphisms. The Ring of integers modulo n, polynomial rings, the Euclidean algorithm.