For the Oral Qualifying Exam
- 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.
- 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.
- 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.
- M. Artin, Algebra
- Dummit and Foote, Algebra
- S. Lang, Algebra
- T.W. Hungerford, Algebra