Galois module structure of algebraic integers of cyclic cubic fields

Abstract

We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-n{X}^2-(n+3)X-1$. Let $L_n=\mathbb{Q}({\rho }_n)$ be a cyclic cubic field with Galois group $G:=\mathrm{Gal}(L_n/\mathbb{Q})$, where ${\rho }_n$ is a root of $f_n(X)$, and ${\mathcal{O}}_{L_n}$ the ring of integers of $L_n$. We explicitly give the generator of the free module ${\mathcal{O}}_{L_n}$ of rank $1$ over the associated order ${\mathcal{A}}_{L_n/\mathbb{Q}}:=\{x\in \mathbb{Q}[G]|x{\mathcal{O}}_{L_n}\subset {\mathcal{O}}_{L_n}\}$ by using the roots of $f_n(X)$.