- Galois theory.
- Homological algebra.
- Representation theory.
I. Galois Theory
- Fields, field extensions, Galois groups.
- Fundamental theorem of Galois theory.
- Cyclotomic and abelian extensions.
- Finite fields.
- Transcendental extensions.
Optional: Hilbert’s theorem 90.
II. Homological Algebra
- Complexes, injective and projective resolutions.
- Derived functors, homology and cohomology.
- Ext and Tor and relations to extensions.
Optional: Group cohomology.
III. Representation Theory
- Basic definitions of representation theory of algebras, particularly modules over group rings.
- Schur’s lemma, character theory and Schur orthogonality relations.
- 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).