Galois structure on integral valued polynomials

TL;DR

This paper characterizes finite Galois extensions of the rationals using rings of integral-valued polynomials and explores constructing bases for these rings as modules over the integers.

Contribution

It provides a characterization of Galois extensions via rings of integral-valued polynomials and addresses basis construction for these rings as Z-modules.

Findings

Characterization of Galois extensions using polynomial rings

Construction methods for bases of polynomial rings as Z-modules

Insights into the structure of integral-valued polynomial rings

Abstract

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.