Fields: basic properties and examples, the characteristic, prime fields
Polynomials: irreducibility, the Eisenstein criterion, Gauss’s lemma
Extensions of fields: the tower property, algebraic and transcendental extensions, adjoining an element to a field
Ruler and compass constructions
Algebraic closures: existence and uniqueness
Splitting fields
Galois extensions: automorphisms, normality, separability, fixed fields, Galois groups, the fundamental theorem of Galois theory.
Cyclic extensions
Solving polynomial equations by radicals: the Galois group of a polynomial, the discriminant, the Cardano-Tartaglia method, solvable groups, Galois theorem
Roots of unity: cyclotomic fields, the cyclotomic polynomials and their irreducibility
Finite fields: existence and uniqueness, Galois groups over finite fields, primitive elements