Groups, rings, fields, homomorphisms, and quotient structures. (Offered fall semester only.)

MATH 261 Linear Algebra

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.

John B. Fraleigh, A First Course in Abstract Algebra, 7^th edition, Pearson, 2003 (ISBN: 9780201763904)

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

Juniors and seniors majoring in Mathematics

1. To gain an introduction to groups and rings.
2. Mastery of computational details in algebraic structures is foundational to the work.
3. Theoretical concepts will be central to cover the major theorems and their proofs.
4. Deductive proof is the distinctive reasoning and writing form of the mathematical
   discipline.
5. Emphasis on deductive proof and problem solving with complete and
   clear written presentation of solutions.

1. Binary operations and isomorphic structures
2. Definitions of a group, examples
3. Subgroups, cyclic groups
4. Permutation groups, orbits, cycles and alternating groups
5. Cosets, Lagrange’s theorem
6. Direct products of groups
7. Homomorphisms, factor groups
   
8. Rings and fields, definitions and examples
   
9. Integral Domains
   
10. Fermat’s and Euler’s theorems
    
11. Quotient fields
    
12. Polynomial rings
    
13. Homomorphisms, factor rings, ideals