Groups, rings, fields, homomorphisms, and quotient structures. Prerequisite: MATH 261. (Offered fall semester.)

John B. Fraleigh, A First Course in Abstract Algebra, 7^th edition, Addison Wesley, Boston, New York, 2002.

Angela C. Hare, Ph.D., Associate Professor of Mathematics

Juniors and seniors majoring in Mathematics.

1. To gain a good working knowledge of the standard basic terminology, definitions and techniques of modern algebra.
2. To have an in depth encounter with axiomatic mathematics with attention to concise definitions and properly constructed proofs.
3. To develop skill in concise, clear and logical presentation of problem solutions and proofs both orally and in writing.
4. To experience the interplay between attempts to find a proof and attempts to find a counterexample which eventually culminates in determination of truth or falsehood of a conjecture.
5. To carry out some of the significant computational exercises of this branch of mathematics, for example construction of the multiplication table for a finite field.
6. To gain familiarity with the use of the program Exploring Small Groups.

1. Abstract vector spaces.
2. Function concept including real-valued function, linear transformations of vector spaces, determinant as a mapping, etc.
3. Polynomials and rational root test.

1. Groups: definition, subgroups, order, permutation groups, Lagrange's theorem, homomorphisms, normal subgroups, quotient (factor) groups, direct product of groups.
2. Rings: definition, subrings, integral domains, ideals and factor rings, direst products, homomorphisms, quotient rings, rings of polynomials, factorization.
3. Fields: definition, extension fields, finite fields.

1. Classrooms with PC access to network.
2. Computer labs (Frey 145, 166) for student use of Exploring Small Groups software.

This course focuses on the deductive method and mathematical proof. The proof writing exercises are a distinguishing feature.