Abstract Algebra II –
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 Advisor.
Foundations of Rings
- 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.
- Apply the fundamental theorem of algebra.
- Explain properties of polynomials.
- Use the division algorithm to divide polynomials.
- Use the factorization process.
- Apply the properties of unique factorization domains as a generalization of polynomials and integers.
- 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
- Analyze a simple extension of a field.
- Calculate the degree of a field extension.
- Apply basic elements and theorems of splitting fields.
- 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.
- Identify properties of separable polynomials and normal extensions.
- Apply Galois theory in fields and polynomials.
- Assess the relationship between solvability of polynomials by radicals and properties of Galois groups.
- The University of Phoenix reserves the right to modify courses.
- While widely available, not all programs are available in all locations or in both online and on-campus formats. Please check with a University Enrollment Advisor.
- Transferability of credit is at the discretion of the receiving institution. It is the student’s responsibility to confirm whether or not credits earned at University of Phoenix will be accepted by another institution of the student’s choice.