Show that is irreducible over . Use the fact (to be proven later) that is a root of to argue that the regular 7-gon is not constructible by straightedge and compass.
Using the rational root theorem, any rational roots of must be either 1 or -1. Certainly then has no rational roots. Since has degree 3, it is irreducible over if and only if it has no roots (in ), so that is irreducible. In particular, the roots of lie in degree 3 extensions of .
Suppose now that the regular 7-gon is constructible. The exterior angles of a regular 7-gon have measure , so in particular, if the regular 7-gon is constructible, then so is the point , so that the number is constructible. But we have seen that any constructible element must have degree a power of 2 over . Given that is a root of , this yields a contradiction.