Abstract Algebra Qualifying Exam Syllabus

Kent Sate University

Department of Mathematical Sciences

Groups

Including homomorphism theorems, permutation groups, automorphisms, finitely generated abelian groups, products of groups, group actions, Sylow theorems, p-groups, nilpotent groups, solvable groups, normal and subnormal series, Jordan-Holder Theorem, special subgroups (e.g. commutator subgroup, Frattini subgroup, etc.).

Rings

Including matrix rings, polynomial rings, factor rings, endomorphism rings, rings of fractions, localization and local rings, prime ideals, maximal ideals, primary ideals, integral domains, Euclidean domains, principal ideal rings, unique factorization domains, Jacobson radical, chain conditions, modules, factor modules, irreducible modules, Artinian and Noetherian rings and modules, semisimplicity.

Fields

Including algebraic extensions, algebraic closures, normal extensions and splitting fields, separable and purely inseparable extensions, theorem of the primitive element, Galois theory, finite fields, cyclotomic extensions, cyclic extensions, radical extensions and solvability by radicals, transcendental extensions.

Linear Algebra

Including matrix theory, eigenvalues and eigenvectors, characteristic and minimal polynomials, diagonalization, canonical forms, linear transformations, vector spaces, bilinear forms, inner products, inner product spaces, duality, tensors.

Suggested References

D. Dummit and R. Foote -- Abstract Algebra
I.M. Isaacs -- Algebra - A Graduate Course
T. Hungerford -- Algebra
N. Jacobson -- Lectures in Abstract Algebra, Vols. I, II, III
N. Jacobson -- Basic Algebra I and II
MacLane and Birkhoff -- Algebra
S. Lang -- Algebra
M. Hall -- The Theory of Groups
J. Rose -- A Course on Group Theory
J. Rotman -- The Theory of Groups, An Introduction
E. Artin -- Galois Theory
Hoffman and Kunze -- Linear Algebra

This document is also available as a PDF file.

Donald White

white@math.kent.edu