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