commutative rings: Noetherian rings and modules, Hilbert Basis
Theorem, Krull dimension, primary decomposition, integral extensions,
integral closure, lying over theorem, Artin rings, Dedekind domains,
discrete valuation rings;
field theory: degree of an extension, transcendence
degree, algebraic closures, separable extensions, splitting fields, normal
extensions, finite fields, purely inseparable
extensions, Primitive Element Theorem, perfect fields;
Galois theory:
Galois extensions, fixed fields,
Galois groups, Fundamental Theorem of Galois Theory, Kummer's theorem,
linear independence of characters, Hilbert's Theorem 90, algebraic
independence of automorphisms, Artin-Schreier theorem, normal basis
theorem, geometric constructibility, solvability by radicals;
non-commutative rings: Wedderburn's theorem on finite division
rings, Frobenius's theorem on division algebras over the reals,
Wedderburn's theorem on simple algebras over a field.
There are regular problem sets (about one every week and a half), and a
final exam on Monday, May 7, from 11am-1pm.