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.
Assess the relationship between solvability of polynomials by radicals and properties of Galois groups.
Apply Galois theory in fields and polynomials.
Identify properties of separable polynomials and normal extensions.
Determine the correspondence between the set of all subgroups of the Galois group and the set of all subfields of the splitting field.
Foundations of Rings
Find complex roots of unity for any natural number n.
Apply the fundamental theorem of algebra.
Determine if a mapping is an isomorphism of rings or fields.
Prove whether a set with specified operations forms a field or ring.
Prove whether a ring is an integral domain.
Galois Theory: Overview
Detect the relationship between powers of prime numbers and the order of a finite field.
Apply basic elements and theorems of splitting fields.
Calculate the degree of a field extension.
Analyze a simple extension of a field.
Detect the relationships between Euclidean, principal ideal, unique factorization, and integral domains.
Determine elements of F[x]/I, where F is a field and I is the ideal (p(x)).
Apply the fundamental homomorphism theorem for rings.
Decide if a mapping is a homomorphism of rings.
Apply the properties of unique factorization domains as a generalization of polynomials and integers.
Use the factorization process.
Use the division algorithm to divide polynomials.
Explain properties of polynomials.
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