Galois module structure of algebraic integers of the simplest cubic field

Let $L_{n}$ be a simplest cubic field with Galois group $G=\mathrm{Gal}(L_{n}/\mathbf{Q})$, $\mathcal{O}_{L_{n}}$ the ring of integers of $L_{n}$. Leopoldt showed that $\mathcal{O}_{L_{n}} \simeq \mathcal{A}_{L_{n}/\mathbf{Q}}$ as $\mathcal{A}_{L_{n}/\mathbf{Q}}$-modules, where $\mathcal{A}_{L_{n}/\mathbf{Q}}$ is the associated order for $L_{n}/\mathbf{Q}$. In this paper, we give a generator of the $\mathcal{A}_{L_{n}/\mathbf{Q}}$-module $\mathcal{O}_{L_{n}}$ explicitly using the roots of Shanks’ cubic polynomial.