The Galois correspondence between the set of subfields of a finite Galois extension and the set of subgroups of the Galois group converts problems about roots of a separable polynomial to problems about the Galois group of its splitting field. We shall see that difficult problems about polynomials are converted into much simpler problems about finite groups.

The Galois correspondence is perhaps the first example of a well-established technique in mathematics: find a suitable formulation for a problem in one branch of mathematics in another branch where the problem becomes much easier to solve.

We will see that the Galois correspondence is powerful enough to provide new ways to prove old results and solve new problems as well. This will be demonstrated here by giving a new proof of the fundamental theorem of algebra. We will also finish the proof of Gauss' criterion for constructibility of regular polygons. We shall derive an expression for in terms of square roots which proves that a seventeen sided regular polygon is constructible by ruler and compass.

We will provide concrete examples of Galois correspondence for some polynomials. In later sections we will derive formulas for the roots of cubic and quartic polynomials as a consequence of the Galois correspondence. Let us begin by proving: