620-321 Algebra

Credit Points

HECS Band

Coordinator

Prerequisites

620-222 (grade of H3 or better).

Semester

Contact

Subject Description

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: abstract rings and isomorphisms; examples including 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; integral domains and the field of quotients; Euclidean domains and principal ideal domains. Modules: submodules, homomorphisms of modules, quotient modules; free modules and bases; structure of a finitely generated module over a principal ideal domain; applications to abelian groups and to Jordan normal form of matrices. Field Theory: field extensions and their construction; the degree of a field extension. Applications, which may include: tensor and exterior algebras, applications to number theory, the classical impossibility theorems, structure theory for simple rings.

Assessment

Up to 24 pages of written assignments and a 3-hour end-of-semester written examination.