Group Theory
• Subgroups, Lagrange's theorem
• Normal subgroups, quotient groups
• Isomorphism theorems, permutation groups, simplicity of A_n
• Cyclic groups, direct products (sums)
• Finitely-generated Abelian groups, p-groups, Sylow theorems

Vector Spaces and Modules
• Submodules, quotient modules, isomorphism theorems
• Linear independence, bases, linear operators, homomorphisms
• Rank, determinant, finitely-generated modules over PID's
• Bilinear and quadratic forms
• Inner product spaces, orthogonality (Gram-Schmidt)
• Dual spaces, determinants, characteristic & minimal polynomials
• Eigenvalues and eigenvectors, Cayley-Hamilton 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

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