Abstract Algebra Qualifying Exams
Kent Sate University
Department of Mathematical
Sciences
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
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D. Dummit and R. Foote -- Abstract Algebra
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I.M. Isaacs -- Algebra - A Graduate Course
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T. Hungerford -- Algebra
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N. Jacobson -- Lectures in Abstract Algebra, Vols. I, II, III
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N. Jacobson -- Basic Algebra I and II
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MacLane and Birkhoff -- Algebra
-
S. Lang -- Algebra
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M. Hall -- The Theory of Groups
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J. Rose -- A Course on Group Theory
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J. Rotman -- The Theory of Groups, An Introduction
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E. Artin -- Galois Theory
-
Hoffman and Kunze -- Linear Algebra
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.
Previous Exams
The table below lists links to all previous algebra qualifying exams given
since August 1991.
Donald White
white@math.kent.edu