Inteiros algébricos e a questão da comensurabilidade entre o lado e as diagonais de um polígono regular

Abstract

In this article, we present a study on the commensurability between the diagonals and the side of regular polygons. Beginning by providing a formula for the ratio between the lengths of the shortest k-th diagonal and the side of a regular n-sided polygon, we will reduce the study of the question of the incommensurability between the k-th shortest diagonal and the side of a regular polygon of n sides to the decision of the rationality or irrationality of this reason. For this, we will use the algebraic integers.