Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1.
Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID.
Let p be a rational prime, p doesn't divide disc(K). Then p is splits completely or inert in K.
Consider the unit group $\mathcal{O}_K^*$. Does there exists a unit $\epsilon$ in $\mathcal{O}_K^*$ such that $(\pi)$ | $(\epsilon -1)$ but $(\pi^2)$ does not divide $(\epsilon -1)$ where $\pi$ is a unramified prime over $\mathbb{Q}$?