Abstract Algebra Qualifying Exams

Kent Sate University

Department of Mathematical Sciences

Time Remaining Until August 2008 Qualifying Exams:

Syllabus

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 DVI, Postscript, or PDF file.

Problem List

The table below lists links to files containing algebra qualifying exam problems. The Complete List contains all problems from all areas, or you can choose the sublist of problems in any of the four main areas. Many, but not necessarily all, of the problems on the algebra qualifying exams will be taken from this list. While we have made an effort to make sure all problems are correct, there may be typographical errors or errors in the statements of some problems.

Please report any access problems or errors in the problem lists to white@math.kent.edu

The lists will be periodically updated and expanded. They were last updated on June 2, 2007.

This update includes minor revisions in notation, deletion of a few problems, and addition of some new problems. The problems marked NEW in the lists have been added or revised (hints, definitions, or parts added) since the previous update on Jaunary 19, 2005.

PDF File
Complete List
Linear Algebra
Group Theory
Ring Theory
Field Theory

Previous Exams

The table below lists links to all previous algebra qualifying exams given since August 1991.

DVI File	Postscript File	PDF File
August 1991	August 1991	August 1991
January 1992	January 1992	January 1992
August 1992	August 1992	August 1992
January 1993	January 1993	January 1993
August 1993	August 1993	August 1993
January 1994	January 1994	January 1994
August 1994	August 1994	August 1994
August 1995	August 1995	August 1995
August 1996	August 1996	August 1996
January 1997	January 1997	January 1997
August 1997	August 1997	August 1997
August 1998	August 1998	August 1998
August 2000	August 2000	August 2000
August 2002	August 2002	August 2002
August 2003	August 2003	August 2003
August 2004	August 2004	August 2004
January 2005	January 2005	January 2005
January 2006	January 2006	January 2006
August 2006	August 2006	August 2006
August 2007	August 2007	August 2007

Donald White

white@math.kent.edu