We characterize finite Galois extensions $K$ of the field of rational numbers in terms of the rings ${ \rm Int}_{ \mathbb{ Q}}(\mathcal O_K)$, recently introduced by Loper and Werner, consisting of those polynomials which have coefficients in $\mathbb{ Q}$ and such that $f(\mathcal O_K)$ is contained in $\mathcal O_K$. We also address the problem of constructing a basis for ${ \rm Int}_{ \mathbb{ Q}}(\mathcal O_K)$ as a $\mathbb{ Z}$-module.
Categories: math.NT math.RA