Galois module structure of algebraic integers of the simplest cubic field

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