## The Syllabus for the Qualifying Examination in Algebra

## 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:

Artin, Algebra

Dummit and Foote, Algebra

Lang, Algebra

Hungerford, Algebra