Constructing regular polygons

In this video, we give a classical application of Galois theory to determine which regular n-gons are constructible by ruler and compass. They are the ones where the order of the unit group of Z/n is a power of two. We start by going through what we mean by ruler and compass constructions and the associated notion of constructible numbers. We then give the classical criterion for when numbers are constructible in terms of towers of quadratic field extensions. Ruler and compass constructions of regular n-gons is then reduced to the Galois theory of the n-th cyclotomic field extension of the rationals. Warning: there is a small error in the first diagram. Can you spot it? It's around the 2:15 mark.