Handbook

Formulate abstract concepts in algebra;

Use a variety of proof-techniques to prove mathematical results;

Work with the most commonly occurring rings and fields: integers, integers modulo n, matrix rings, rationals, real and complex numbers, more general structures such as number fields and algebraic extension fields, splitting fields, algebraic integers and finite fields;

Understand different types of rings, such as integral domains, principal ideal domains, unique factorisation domains, Euclidean domains, fields, skew-fields; amongst these are the Gaussian integers and the quaternions - the best-known skew field;

Apply the classification of finite fields;

Generalise known concepts over the integers to other domains, for example, use the Euclidean algorithm or factorisation algorithms in the algebra of polynomials;

Construct larger fields from smaller fields (field extensions and splitting fields);

Apply field theory to coding and cryptography; understand the classification of cyclic codes.