After a preliminary study of fixed fields of field automorphisms, the fundamental theorems in Galois theory are proved. Having also studied the Galois group of a polynomial, various applications are included, like a proof of the fundamental theorem of algebra, the study of constructable polygons, and of polynomials solvable by radicals. During this, one also considers notions like primitive roots and cyclotomic polynomials, cyclic extensions, and symmetric functions. The final section concerns the classical cubic and quartic equations.