MATH 6320 Algebra

Textbook

Dummit & Foote, Abstract Algebra (in recent years)

Main Topics

  1. Galois theory.
  2. Homological algebra.
  3. Representation theory.

Syllabus

I. Galois Theory

  1. Fields, field extensions, Galois groups.
  2. Fundamental theorem of Galois theory.
  3. Cyclotomic and abelian extensions.
  4. Finite fields.
  5. Transcendental extensions.

Optional: Hilbert’s theorem 90.

II. Homological Algebra

  1. Complexes, injective and projective resolutions.
  2. Derived functors, homology and cohomology.
  3. Ext and Tor and relations to extensions.

Optional: Group cohomology.

III. Representation Theory

  1. Basic definitions of representation theory of algebras, particularly modules over group rings.
  2. Schur’s lemma, character theory and Schur orthogonality relations.
  3. Maschke’s and Wedderburn’s theorems.

Optional: Tensor products of representations, induced modules.  Introduction to representation theory of the symmetric group and SL (2, Fq).