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- 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.
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