Générateurs de l'anneau des entiers d'une extension cyclotomique

Let $p$ be an odd prime and $q = p^m$, where $m$ is a positive integer. Let $zeta_q$ be a $q$th primitive root of $1$ and $mathcal{O}_q$ be the ring of integers of $mathbb{Q}(zeta_q)$. I. Ga'al and L. Robertson showed that if $(h_q^+, p(p-1)/2) = 1$, where $h_q^+$ is the class number of $mathbb{Q}(zeta_q + overline{zeta_q})$, then if $alpha in mathcal{O}_q$ is a generator of $mathcal{O}_q$ either $alpha$ is equal to a conjugate of an integer translate of $zeta_q$ or $alpha + overline{alpha}$ is an odd integer. In this paper we show that we can remove the hypothesis over $h_q^+$. In other words we prove that if $alpha$ is a generator of $mathcal{O}_q$, then either $alpha$ is a conjugate of an integer translate of $zeta_q$ or $alpha + overline{alpha}$ is an odd integer.