1.Background. 2.Factorization of polynomials. 3.Field extension. 4.The Degree of an extension. 5.Ruler and compasses. 6.Transcendental numbers. 7.The idea behing Galois Theory. 8.Normality and separability. 9.Field degree and group orders. 10.Monomorphisms, automorphisms, and normal closures. 11.The Galois correspondence. 12.A specific example. 13.Some group theory. 14.Solution of equations by radicals. 15.The general polynomial equation. 16.Finite fields. 17.Regular polygons. 18.The fundamental theorem of algebra. 19.Later developments. Harder exercises.