We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-nX^2-(n+3)X-1$. Let $L_n=\mathbb Q(\rho_n)$ be a cyclic cubic field with Galois group $G:={ \rm Gal}(L_n/\mathbb Q)$, 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 $f_n(X)$.
Categories: math.NT