Field, characteristic of a field, subfield and prime field of a field, field extension, the degree of a field extension, Adjunction of roots, splitting field, finite fields, Algebraically closed fields, existence of algebraic closure, algebraically closed fields. Separable, normal and purely inseparable extensions. Perfect fields, primitive elements. Langrange’s theorem on primitive elements.
Galois extensions, the fundamental theorem of Galois theory, Cyclotomic extensions and Cyclic extensions, Applications of cyclotomic extensions and Galois theory to the constructability of regular polygons, Solvability of polynomials by radicals.