Fields

• Polynomials over a field – irreducibility, Gauss’s lemma, characteristic, perfect fields, cyclotomic polynomials.

• Field extensions – degrees, towers, minimal polynomials, quadratic extensions.

• Splitting fields and normal extensions.

• Separability – separable polynomials and extensions, perfect fields, inseparable extensions in characteristic \(p\).

• Finite fields – existence, uniqueness, subfields, and cyclic Galois groups.

• Fields review sheet 1.

The Galois correspondence is in Galois theory.