#### 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

• 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