Abstract Algebra Qualifying Exams

Kent Sate University

Department of Mathematical Sciences




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.).


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.


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

This document is also available as a 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 September 7, 2023.

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 are the problems that have not appeared on previous exams and have been added since the January 13, 2020 update.

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.
August 1991 January 1992
August 1992 January 1993
August 1993 January 1994
August 1994  
August 1995  
August 1996 January 1997
August 1997  
August 1998  
August 2000  
August 2002  
August 2003  
August 2004 January 2005
  January 2006
August 2006  
August 2007  
August 2008  
August 2009 January 2010
August 2010 January 2011
August 2011 January 2012
August 2012 January 2013
August 2013  
August 2014 January 2015
August 2015 January 2016
August 2016 January 2017
August 2017 January 2018
August 2018 January 2019
August 2019 January 2020
August 2020 January 2021
August 2021 January 2022
August 2022 January 2023
August 2023

Donald White