Algebra

Algebra Syllabus

For the Oral Qualifying Exam

Groups

  • Elementary concepts (homomorphism, subgroup, coset, normal subgroup), solvable groups, commutator subgroup, Sylow theorems, structure of finitely generated Abelian groups.
  • Symmetric, alternating, dihedral, and general linear groups.

Rings

  • Commutative rings and ideals (principal, prime, maximal).
  • Integral domains, Euclidean domains, principal ideal domains, polynomial rings, Eisenstein's irreduciblility criterion, Chinese remainder theorem.
  • Structure of finitely generated modules over a prinicpal ideal domain.

Fields

  • Extensions: finite, algebraic, separable, inseparable, transcendental, splitting field of a polynomial, primitive element theorem, algebraic closure.
  • Finite Galois extensions and the Galois correspondence between subgroups of the Galois group and subextensions.
  • Solvable extensions and solving equations by radicals.
  • Finite fields.

References

  • M. Artin, Algebra
  • Dummit and Foote, Algebra
  • S. Lang, Algebra
  • T.W. Hungerford, Algebra