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.
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.
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.
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