MATH 6320 - Algebra

David Zywina, spring 2016.

MATH 6320 is the second of the two core algebra courses. It treats Galois theory, representation theory of finite groups and associative algebras, and an introduction to homological algebra. For the most part these subjects are not covered in depth, since the purpose of the course is to present a broad view with a glimpse of several topics.

Some topics to be covered

Field theory and Galois theory: Field extensions, degree, splitting fields, algebraic closure, normal and separable extensions, fundamental theorem of Galois theory, solvability of equations by radicals, cyclotomic extensions, finite fields.

Homological algebra: Exact sequences, projective and injective modules, Schanuel's lemma, homological dimension, complexes, homology.

Representation theory of finite groups: Simple and semi-simple rings and modules, Wedderburn's theorem, group representations, Maschke's theorem, characters of finite groups, orthogonality relations, Frobenius reciprocity, applications to group theory.

Textbook

Dummit & Foote, Abstract Algebra