mth402 | undergraduate

Abstract Algebra II

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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

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    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.
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