Group Theory

Subgroups, Lagrange's theorem

Normal subgroups, quotient groups

Isomorphism theorems, permutation groups, simplicity of A_{n}

Cyclic groups, direct products (sums)

Finitelygenerated Abelian groups, pgroups, Sylow theorems
Vector Spaces and Modules

Submodules, quotient modules, isomorphism theorems

Linear independence, bases, linear operators, homomorphisms

Rank, determinant, finitelygenerated modules over PID's

Bilinear and quadratic forms

Inner product spaces, orthogonality (GramSchmidt)

Dual spaces, determinants, characteristic & minimal polynomials

Eigenvalues and eigenvectors, CayleyHamilton theorem

Canonical forms (triangular, rational, Jordan)
Rings

Subrings, ideals, quotient rings, isomorphism theorems

Arithmetic of Z and Z_{n} (Fermat's theorem, Chinese Remainder theorem)

Integral domains and quotient fields

Prime and maximal ideals, euclidean rings, PID's and UFD's

Polynomial rings, Gauss' lemma
Fields

Finite and algebraic extentions, Galois extensions

Simple extensions, finite fields

Galois theory (in characteristic 0), geometric constructions

Solvability by radicals
References
Birkhoff & MacLane: A Survey of Modern Algebra
Fraleigh: A First Course in Abstract Algebra
Herstein: Topics in Algebra
Hungerford: Algebra