Field Theory . Algebraic extensions: degree, minimal polynomials, adjoining a root. Existence and uniqueness of splitting fields, algebraic closure. Finite fields: classification, Frobenius automorphism, cyclicity of finite multiplicative subgroup of a field. Normal extensions, separable closures, perfect fields, primitive element theorem. Inseparable extensions. Galois theory: field embeddings and Galois groups, examples, fundamental theorem in finite case, cyclotomic extensions, norms and traces. Kummer theory, solvability by radicals via solvable groups. Examples. Infinite Galois theory: analogy with the fundamental group of a topological space, Krull topology, fundamental theorem.