On the fundamental units of some cubic orders generated by units

Jun Ho Lee ¹   ; Stéphane R. Louboutin ²

¹ School of Mathematics Korea Institute for Advanced Study (KIAS) 85 Hoegiro, Dongdaemun-gu Seoul 130-722, Republic of Korea
² Aix Marseille Université CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille, France

Acta Arithmetica, Tome 165 (2014) no. 3, pp. 283-299

Résumé

Let $\epsilon $ be a totally real cubic algebraic unit. Assume that the cubic number field ${\mathbb Q}(\epsilon )$ is Galois. Let $\epsilon $, $\epsilon '$ and $\epsilon ''$ be the three real conjugates of $\epsilon $. We tackle the problem of whether $\{\epsilon ,\epsilon '\}$ is a system of fundamental units of the cubic order ${\mathbb Z}[\epsilon ,\epsilon ',\epsilon '']$. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic polynomials are considered. We also improve upon and correct several previous results in the literature.

Détail

DOI : 10.4064/aa165-3-7

Keywords: epsilon totally real cubic algebraic unit assume cubic number field mathbb epsilon galois epsilon epsilon epsilon three real conjugates epsilon tackle problem whether epsilon epsilon system fundamental units cubic order mathbb epsilon epsilon epsilon given units totally real cubic order explain prove form system fundamental units order several explicit families totally real cubic orders defined parametrized families cubic polynomials considered improve correct several previous results literature