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.

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.

Polynomials

  • 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.

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

  • 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.

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