**Field theory**: Field extensions, splitting fields, separable and normal extensions, finite fields, cyclotomic fields.

**Galois theory**: Fundamental theorem of Galois theory, solvability by radicals, computing some Galois groups.

**Topics from Algebraic number theory**: fractional ideals, unique factorization in a Dedekind domain.

**Topics from Algebraic geometry**: Hilbert basis theorem, nullstellensatz, tensor products.

**Topics from Homological Algebra**: categories, functors, exact sequences, projective and injective modules, flatness, direct and inverse limits.