The preliminary examination in algebra covers topics in group theory; ring theory; module theory; and field theory and Galois theory. Most, and usually all, the required topics are discussed annually in Mathematics 551-552, and can also be found in “Abstract Algebra” by Dummit and Foote as well as “Algebra” by Lang or “Algebra” by Hungerford.

In addition to the specific items listed below, students should know the basic concepts applicable to several areas (such as homomorphism, kernel, isomorphism theorems, generating sets, direct product, universal mapping property, center, etc.).

Group Theory: basic examples (dihedral and quaternion groups, symmetric groups, alternating groups, group of invertible elements of ring, matrix groups, automorphism groups), cyclic groups, finitely generated abelian groups, free abelian groups, types of subgroups (normal, commutator, normalizer, Sylow), Cayley’s theorem, simple groups, composition series, Jordan-Hölder theorem, finite solvable groups, group actions, stabilizers, orbits, class equation, p-groups, Sylow Theorems.

Ring Theory: prime and maximal ideals, Chinese Remainder theorems, commutativity, radical of an ideal, nilradical, Jacobson radical, zero-divisor, integral domain, division ring, field, local ring, polynomial rings, division algorithm, Euclidean domain, principal ideal domain, unique factorization domain, noetherian ring, localization, quotient fields, primitive polynomials and the lemma of Gauss, Eisenstein’s criterion.

Module Theory: exact sequences, the Hom functors, basis, free module, projective modules, tensor product of modules and of homomorphisms, chain conditions on modules, structure of finitely generated modules over a principal ideal domain (including the fundamental theorem of abelian groups).

Field Theory and Galois Theory: prime fields, characteristic, algebraic extensions, separable extensions, purely inseparable extensions, splitting fields, algebraic closures, Galois extensions, fundamental theorem of Galois theory, computation of Galois groups, structure of finite fields, perfect fields, primitive element theorem, cyclotomic extensions, solvability by radicals.

Reviewed: 2025