620-321 Algebra

620-222 with a grade of H3 or better.

This subject provides further experience with abstract algebraic concepts and methods. General structural results are proved and algorithms developed to determine the invariants they describe. The material covered is widely used in algebraic topology and in number theory.

Rings topics include: abstract rings and isomorphisms; matrix rings and polynomial rings; homomorphisms, ideals and quotient rings; integral domains and the field of quotients; units, irreducibles and primes; prime and maximal ideals; Euclidean domains; principal ideal domains; and unique factorisation domains. Modules topics include: submodules; homomorphisms of modules and quotient modules; free modules and bases; the structure of a finitely generated module over a principal ideal domain; and applications to abelian groups and to Jordan normal form of matrices. Field theory topics include: field extensions and their construction; the degree of a field extension; Galois extensions, splitting fields and the Galois correspondence. Applications topics may include tensor and exterior algebras, applications to number theory, the classical impossibility theorems, and structure theory for simple rings.

Up to 24 pages of written assignments due during semester (20%); a 3-hour written examination in the examination period (80%).