| I. Modules |
| modules, submodules and quotients, homomorphisms, basis and dimension, direct products and direct sums, free modules, projective modules, Modules over PIDs, torsion and torsion free modules, primary modules, structure theorem, rational canonical form, exact sequences, 5-lemma, snake lemma, tensor product of modules, flat modules, injective modules, localization, Nakayama's lemma, locally free modules, Ext and Tor, limits. |
| II. Commutative Rings |
| Noetherian modules, Artinian modules, Krull's intersection theorem, integrality and finiteness, going up, down, lying over, dimension, height, Artinian rings, Krull's principal ideal theorem, regular local rings, Dedekind domains, discrete valuation rings, (fractional ideals, Cartier divisors, ideal class group). |
| III. Fields and Field Extensions |
| prime fields, algebraic and transcendental elements, finite extensions, algebraic extensions, transendental extensions, stem fields, splitting fields, algebraic closure, automorphisms, normal extensions, finite fields, resultants, discriminant, separable polynomials, perfect fields, separable degree, primitive element theorem. |
| IV. Galois Theory |
| Galois extensions, fixed fields, fundamental theorem of Galois theory, translation theorem, Galois group of a polynomial, roots of unity, cyclotomic fields, Wedderburn's theorem, solvability by radicals, radical extensions, Cardano's formulae, constructions with compass and straight edge, 2-radical extensions, construction of the regular n-gon, invariants of permutation groups, (reduction criterion), normal basis theorem, Kummer and Artin-Schreier extensions, (Galois descent, transcendental extensions). |