Course Descriptions

General Information (Catalog listing)

01:640:451-452 Abstract Algebra I,II (3,3)
Rigorous study of abstract algebraic systems including groups, rings, and fields.
Prerequisites: Permission of department and instructor. For students preparing for graduate study in the mathematical sciences.

Math 452 is part of the Honors Track sequence. It is intended to be taken immediately after Math 451. Students enrolled in Math 451 during the Fall semester should be able to preregister for Math 452 in the Spring without assistance. All other students should contact the Honors Advisor <>.

Required Textbook

Michael Artin; Algebra (second edition); Prentice-Hall, 2011 (560 pp.); (ISBN: 0-13-241377-9; ISBN13: 9780132413770).
See also the Master Textbook List page.

Short Description

Review of vector spaces, linear transformations and matrices, including Jordan canonical form.
Factoring in commutative rings: principal ideal domains, unique factorization domains; examples.
Modules; free modules; diagonalizing integer matrices; application to the structure of finitelygenerated abelian groups.
Modules over polynomial rings; application to rational canonical form for linear operators.

Topics, in approximate order

Review of vector spaces, linear transformations, matrices and Jordan canonical form
Rings
Commutative rings
Homomorphisms and ideals
Quotient rings, product rings
Rings of fractions
Maximal ideals
Factoring in commutative rings
Principal ideal domains
Unique factorization domains
Modules
Free modules
Noetherian rings
Structure of abelian groups
Application to linear operators, rational canonical form

 As time allows, with the instructor’s discretion, some additional topics may also be covered, such as the following:
Quadratic number fields, fields and Galois theory, non-commutative rings, algebras.

Sample Syllabus

Spring 2013

Schedule of Sections

Previous semesters