Xavier Mbaale

Among the ancient Greeks, the perfect geometric figures were the circle and straight line. As such,
they restricted performing geometrical constructions to two instruments: the ruler and compass. In
this essay, we discuss which regular polygons are constructible by ruler and compass. For instance, a regular p−gon for p a prime is constructible if p is of the form 2 ^(2^n) + 1. These primes are known as Fermat primes. But before the development of field theory, Gauss had explicitly constructed a 17−gon. Therefore as an explicit example of a constructible polygon, we shall describe Gauss’s construction of a
17−gon and explain how it works using Galois theory.