# Abstract Algebra II

### Explore by:

or call us at
or call us at

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 To enroll, speak with an Enrollment Representative.

#### Course details:

Credits: 3
Continuing education units: XX
Professional development units: XX
Duration: 5 weeks

#### Polynomials

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

#### Quotient Rings

• Decide if a mapping is a homomorphism of rings.
• Apply the fundamental homomorphism theorem for 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.

#### 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.

#### 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.
Tuition for individual courses varies. For more information, please call or chat live with an Enrollment Representative.