Algebra Preliminary Exam Syllabus
In general, the syllabus is determined by the material covered in
the course MATH 5125/6, Abstract Algebra, in the year which the
student takes the course. This means that the syllabus will depend on
the instructor teaching the course that year;
however, the following topics are nearly always examined.
Groups
- Fundamental isomorphism theorem
- Groups acting on sets
- Alternating and symmetric groups
- Sylow Theorems
- Finite simple groups
- Finitely generated abelian groups
Rings and Fields
- Polynomial rings
- PID's and UFD's
- Prime and maximal ideals
- Field extensions, normal and Galois extensions, separability
- Computing Galois groups
Modules
- Isomorphism theorems
- Bases, direct sums, free modules, simple modules
- Modules over a PID
Additional topics
may include:
- Nilpotent and solvable groups
- Free groups, generators and relations
- Noetherian rings and modules
- Hilbert Basis Theorem
- Hilbert Nullstellensatz
- Power series rings
- Projective and injective modules
- Nakayama's lemma
- Jacobson radical
- Artinian rings and modules
- Semisimple rings and Wedderburn structure theorem
- Jordan and rational canonical forms
- Representation theory of finite groups and character theory
- Tensor products
Books
The following are appropriate for Math 5125/5126.
- ``Algebra" by Michael Artin,
Prentice Hall, 1991, ISBN
0-13-004763-5, MR:
92g:00001
- ``Abstract
Algebra" (third edition)
by David S. Dummit and Richard M. Foote,
Wiley 2003, ISBN 0-471-43334-9 MR2286236
(2007h:00003) and MR: 92k:00007
- ``Algebra" by Thomas W. Hungerford, Springer-Verlag, 1980,
ISBN 0-387-90518-9, MR: 50 #6693
- ``Algebra, a graduate course" by I. Martin Isaacs,
Brooks/Cole,
1994, ISBN 0-534-19002-2, MR: 95k:00003
- ``Basic Algebra" volumes I and II (second editions)
by Nathan Jacobson, Freeman, 1985 and 1989, ISBN 0-7167-1480-9 and
0-7167-1079-X, MR: 86d:00001
and MR: 90m:00007
- ``Algebra" (third edition) by Serge Lang, Addison Wesley, 1993, ISBN 0-201-55540-9
- ``Advanced Modern Algebra" by Joseph J. Rotman
Prentice Hall ISBN 0-13-087868-5
- ``Basic
Algebra and Advanced Algebra Set" by Anthony W.
Knapp, Birkhauser ISBN
0817645330; perhaps too sophisticated for this level, but otherwise
two superb books by a great expositor.
Other Books
The following are good for various
sections of MATH 5125/6, but do not cover the whole syllabus.
- ``Algebra, a Module Theoretic Approach" by William A. Adkins
and Steven H. Weintraub, Graduate Texts in Mathematics no. 136,
Springer-Verlag, 1992,
ISBN 0-387-97839-9, MR 94a:00001
- ``Introduction to Commutative Algebra" by M. F. Atiyah and
I. G. Macdonald,
Addison Wesley, 1969,
MR: 39 #4129
- ``Undergraduate Commutative Algebra" by Miles Reid, London Math.
Soc. Student Texts no. 29, Cambridge University Press, 1995, ISBN 0-521-45889-7,
MR
98c:13001
- `` The Theory of Groups" by Ian D. Macdonald, Krieger Pulblishing
Co., 1988, ISBN 0-89464-287-1, MR
39# 5670
- ``An Introduction to the Theory of Groups" by Joseph J. Rotman,
Graduate Texts in Mathematics no. 148,
Springer-Verlag, 1995, ISBN
0-387-94285-8, MR: 95m:20001
- ``Rings, Modules and Linear Algebra" by Brian Hartley and
Trevor
O. Hawkes, Chapman & Hall, 1980, ISBN 0-412-09810-5, MR: 42 #2897
- ``Field theory and its classical problems" by Charles Robert
Hadlock, Carus Mathematical Monographs no. 19,
Mathematical Association of America,
1978, ISBN 0-88385-020-6, MR: 82c:12001
- ``A First Course in Abstract Algebra" by Joseph J. Rotman,
Prentice Hall, 2000,
ISBN 0-13-011584-3
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Revision Date:
Fri Nov 23 18:06:50 EST 2007