Abstract Algebra II
Revised: September, 2014
Course Description
This course introduces rings including integral domains, ideals, ring homomorphisms
and 铿乪lds including extension 铿乪lds, 铿乶ite 铿乪lds, culminating in the Galois Theory.
Prerequisite: Math 361. Three semester hours.
Student Learning Objectives
By the end of the course students will be able to:
- Know all relevant definitions and correct statements of major theorem;
- Compute Aut(G) for a given group G;
- Use the Sylow Theorems to characterize certain finite groups;
- Compute direct product of groups;
- State and apply the Fundamental Theorem of Finite Abelian Groups;
- Understand the structure of quotient groups and rings, rings of polynomials and field
extensions;
- Use the Isomorphism Theorems to deduce structure and properties of rings and groups;
- Use modern algebra in solving problems such as the impossibility of certain ruler
and compass constructions, and the impossibility of a general formula for roots of
polynomials;
- Applying theory of quotient rings to create extension fields which contain roots of
polynomials which were not present in the base field; and
- Understand the Fundamental Theorem of Galois Theory and its relation to solubility
of polynomial equations.
Text
Topics in Algebra by I.N. Herstein, Second Edition
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 1: Homomorphisms, Automorphisms, Group Theory (Sylow's Theorem, Direct Products,
Finite Abelian Groups) (12 days)
- Chapter 2: Rings (Definition and Examples, Homomorphisms, Ideals and Quotient Rings,
Integral Domain, Euclidean Rings, Polynomial Rings (12 days)
- Chapter 5: Fields (Extension Fields, Roots of Polynomials, Construction with Ruler
and Compass, Elements of Galois Theory, Solvability by Radicals, Galois Groups over
the Rationals) (12 days)
- Chapter 7: Selected Topics (Finite Fields)