Algebra Syllabus (for MS and Preliminary Exams)

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)


  • 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


  • 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