mth402 | undergraduate

Abstract Algebra II

Explore by:

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
Duration: 5 weeks

topic title goes here

    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.


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

    Please ask about these special rates:

    Teacher Rate: For some courses, special tuition rates are available for current, certified P-12 teachers and administrators. Please speak with an Enrollment Representative today for more details.

    Military Rate: For some courses, special tuition rates are available for active duty military members and their spouses. Please speak with an Enrollment Representative today for more details.

    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 Representative.

    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.