# Abstract Algebra II – mth402 (3 credits)

This is the second course in a two-part course sequence presenting students with the applications of abstract algebraic theories. Students will investigate rings, fields, and the basic theorems of Galois theory.

This undergraduate-level course is 5 weeks. This course is available to take individually or as part of a degree or certificate program. To enroll, speak with an Enrollment Representative.

### Quotient Rings

• Determine elements of F[x]/I, where F is a field and I is the ideal (p(x)).
• Detect the relationships between Euclidean, principal ideal, unique factorization, and integral domains.
• Decide if a mapping is a homomorphism of rings.
• Apply the fundamental homomorphism theorem for rings.

### Galois Theory: Overview

• Calculate the degree of a field extension.
• Apply basic elements and theorems of splitting fields.
• Analyze a simple extension of a field.
• Detect the relationship between powers of prime numbers and the order of a finite field.

### Galois Theory and Geometric Constructions

• Determine the correspondence between the set of all subgroups of the Galois group and the set of all subfields of the splitting field.
• Apply Galois theory in fields and polynomials.
• Identify properties of separable polynomials and normal extensions.
• Assess the relationship between solvability of polynomials by radicals and properties of Galois groups.

### Polynomials

• Use the factorization process.
• Explain properties of polynomials.
• Use the division algorithm to divide polynomials.
• Apply the properties of unique factorization domains as a generalization of polynomials and integers.

### Foundations of Rings

• Apply the fundamental theorem of algebra.
• Prove whether a ring is an integral domain.
• Prove whether a set with specified operations forms a field or ring.
• Determine if a mapping is an isomorphism of rings or fields.
• Find complex roots of unity for any natural number n.