Group Theory
- Subgroups, Lagrange's theorem
- Normal subgroups, quotient groups
- Isomorphism theorems, permutation groups, simplicity of An
- 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 Zn (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
