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}$.

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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.

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