MATH 461 Syllabus

Abstract Algebra II

Revised: September, 2014

Course Description

This course introduces rings including integral domains, ideals, ring homomorphisms and fields including extension fields, finite fields, 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)

Office of Web Services