MATH 6320: Algebra (spring 2010)

Instructor: Martin Kassabov

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.

Prerequisite: Officially, MATH 6310. In practice, a solid understanding of the material of an honors undergraduate course in abstract algebra, such as Cornell's MATH 4340, should suffice.

Text: Dummit & Foote, Abstract Algebra (Edition: 3), John Wiley & Sons, 2004 (ISBN: 0-471-43334-9).

Some topics to be covered:

  1. 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.
  2. Homological algebra: Exact sequences, projective and injective modules, Schanuel's lemma, homological dimension, complexes, homology.
  3. 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.

The course will start with material from chapters 13 and 14 of Dummit and Foote. Students should have a basic understanding of material from chapters 1–12 (e.g., 1–6.1, 7, 8, 9.1–9.5, 10.1–10.4, 11.1–11.4, 12) although certainly not all will be used.