Units in real cyclic fields

Abstract

Let $N/\mathbb{Q}$ be a real cyclic and tame extension of prime degree $l$ with $\Gamma=\mathscr{G}al(N/\mathbb{Q})$. We give the Hom description of the class of the torsion-free part of the group of units in $N$ in the class group of the order $\mathbb{Z}\Gamma/ (\sum_{\gamma \in \Gamma}\gamma )$. This representation depends only on the structure of the ideal class group of $N$ and determines the Galois module structure of the torsion-free part of the group of units in $N$ as an ideal of the $l$th cyclotomic field. Using this approach we derive necessary and sufficient conditions for all real and tame cyclic fields of prime degree to have Minkowski units. We extend also the class of known cyclic real fields with Minkowski units.