Abstract Algebra
Revised: November 2006
Course Description
Topics from abstract algebra including quotient groups and rings, rings of polynomials and field extensions, quaternions, and homomorphism theorems. Prerequisite: Math 361. (Three semester hours)
Objectives
1. To expose the student to some relatively deep theorems in abstract algebra.
2. To further develop the student's ability to reason abstractly.
Text
Joseph A. Gallian, Contemporary Abstract Algebra, Fifth Edition. Houghton Mifflin.
Grading Procedure
Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.
Attendance Policy
Attendance policy is left to the discretion of individual instructors, subject to general university policy.
Course Outline
- Chapter 12: Rings (3 days)
Definition and examples, properties, subrings - Chapter 13: Integral Domains (2 days)
Definition and examples, properties, fields, characteristic of a ring - Chapter 14: Ideals and Factor Rings (2 days)
Ideals, Factor Rings, Prime and Maximal Ideals - Chapter 15: Ring Homomorphisms (3 days)
Definition and examples, properties, ring homomorphisms, field of quotients - Chapter 16: Polynomial Rings (3 days)
Notation and terminology, division algorithm. - Chapter 17: Factorization of Polynomials (2 days)
Reducibility tests, Irreducibility tests, Unique Factorization - Chapter 18: Divisibility in Integral Domains (3 days)
Irreducibles, primes, Unique Factorization Domains, Euclidean Domains - Chapter 19: Vector Spaces ( 3 days)
Definition and examples, subspaces, linear independence - Chapter 20: Extension Fields (3 days)
Fundamental Theorem of Field Theory, Splitting Fields, Zeros of an Irreducible Polynomial. - Chapter 21: Algebraic Extensions (3 days)
Characterization, Properties of Extensions - Chapter 22: Finite Fields (3 days)
Characterization, Structure, Subfields of a Finite Field - Special topics from part 5 as time allows