Gauss i el polígon de 17 costats

Abstract

In this article we highlight the historical importance of the construction of the 17-sided regular polygon, problemat which Gauss devoted the first work that he published.We?ll see how the same procedure followed in this case, which involves grouping the roots of a polynomial of degree 17 − 1 = 16 in two groups of eight roots each, and we divide again each of these groups in two groups, up to arrive to eight groups of two roots each, can be generalized to construct the n-sized regular polygon when n − 1 is a power of two. From this it is easily seen that if in the prime factor decomposition of n it appears only powers of two and Fermat prime numbers, then the regular polygon of n sides can be constructed with ruler and compas. The reciprocal of this theorem, given by Gauss without proof, is known today asWantzel theorem. In an appendix we give a proof of this theoremusing field extensions andminimal polynomials, two of the basic ingredients of Galois theory.