# Math 372: Abstract Algebra 2.

### Title

Abstract Algebra 2.

(3:3:0)

F, W

### Description

Fields, Galois theory, solvability of polynomials by radicals.

## Desired Learning Outcomes

### Prerequisites

It is critical for students to have mastered Math 313 (linear algebra) and Math 371 (group theory and ring theory) prior to enrolling in Math 372.

### Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

1. Ring Theory
• Basic Definitions
• Examples of rings (both commutative and noncommutative)
• Ideals
• Ring homomorphisms
• Quotient rings
• Prime and maximal ideals
• Polynomial rings over fields
• Factorization in polynomial rings
• Irreducible polynomials
• Field of fractions of a domain
2. Field Theory
• Extensions of fields
• Field extensions via quotients in polynomial rings
• Automorphisms of fields
• Field of characteristic 0 and prime characteristic
• Galois extensions and Galois groups
• The Galois correspondence
• Independence of characters
• Fundamental Theorem of Galois Theory
• Fundamental Theorem of Algebra
• Roots of unity
• Insolvability of the quintic

### Textbooks

Possible textbooks for this course include:

Joseph Rotman, Galois Theory (Second Edition), Springer, 1998.

David Dummit and Richard Foote, Abstract Algebra (Third Edition), Wiley, 2003. (The chapters on fields and Galois theory and probably including some of the material on rings.)

Ian Stewart, Galois Theory (Third Edition), Chapman Hall, 2004.