# Gauss and Regular Polygons: Conclusion

### The Central Result

This is the concluding post in this series and we aim to prove the following result (proved in part by Gauss and finally the converse by Wantzel):
A regular polygon of $n, n > 2$ sides can be constructed by Euclidean tools if and only if $\phi(n) = 2^{k}$.

# Gauss and Regular Polygons: Gaussian Periods Contd.

### Properties of Gaussian Periods

In this post we are going to establish the following properties of the Gaussian periods which will ultimately lead to a solution of the equation $z^{p} - 1 = 0$. Again as in previous post, $p$ is to be considered a prime unless otherwise stated. In the following we have $e, f$ as two positive integers with $ef = (p - 1)$.
1. Any period of $f$ terms can be expressed as a polynomial in any other period of $f$ terms with rational coefficients.
2. If $g$ divides $(p - 1)$ and $f$ divides $g$, then any period of $f$ terms is a root of a polynomial equation of degree $g / f$ whose coefficients are rational expressions of a period of $g$ terms.