Galois module structure of algebraic integers of the simplest cubic field

Abstract

Let $L_n$ be a simplest cubic field with Galois group $G=\mathrm{Gal}({\mathrm{L}}_{\mathrm{n}}/\mathbb{Q})$. The associated order is denoted as ${\mathcal{A}}_{L_n/\mathbb{Q}}:=\{x\in \mathbb{Q}[G]|x\cdot {\mathcal{O}}_{{\mathcal{L}}_n}\subset {\mathcal{O}}_{{\mathcal{L}}_n}\}$, where ${\mathcal{O}}_{L_n}$ is the ring of integers of $L_n$. Leopoldt showed that ${\mathcal{O}}_{{\mathcal{L}}_n}\simeq {\mathcal{A}}_{{\mathcal{L}}_n/\mathbb{Q}}$ as ${\mathcal{A}}_{L_n/\mathbb{Q}}$-modules. In this paper, we give a generator of the ${\mathcal{A}}_{L_n/\mathbb{Q}}$-module ${\mathcal{O}}_{L_n}$ explicitly using the roots of Shanks' cubic polynomial. If $L_n/\mathbb{Q}$ is tamely ramified, then we have ${\mathcal{A}}_{L_n/\mathbb{Q}}=\mathbb{Z}[G]$, and the conjugates form a normal integral basis, which has been obtained explicitly in the previous work of Hashimoto and the second author.