  Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers
$\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a
natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed
sequences, equipped with an $\mathcal O_K$-invariant probability measure
associated to $\mathcal F_k$. We prove that this action is ergodic, has pure
point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian
group. In particular, it is not weakly mixing and has zero measure-theoretical
entropy. This work generalizes the paper by the first author and Sinai
arXiv:1112.4691 [math.DS] where $K=\mathbf Q$ and $k=2$.
