ruler and compass construction of regular polygons (thing)

Interestingly, to construct a regular p-gon with p a Fermat prime, the first step is to construct sqrt(p).

Other random facts about constructing polygons with straightedge and compass:

• Gauss requested that his construction of the 17-gon be inscribed on his tombstone (apparently it wasn't).
• The proof of Gauss' theorem stated by Noether (the E2 user, not the Emily Noether!) uses Galois theory.
• Gauss couldn't have known about Galois theory.