Polynomial Index in Dedekind Rings

TL;DR

This paper establishes a lower bound for the p-adic valuation of the index of a polynomial over Dedekind rings, linking it to the factorization of its reduction modulo prime ideals, with implications for power integral bases.

Contribution

It provides a new lower bound for the p-adic valuation of polynomial indices in Dedekind rings based on factorization properties, aiding in understanding power integral bases.

Findings

Lower bound for p-adic valuation in terms of factor degrees

Criterion for non-existence of power integral bases

Application to field extension analysis

Abstract

Let $R$ be a Dedekind ring, $\mathfrak{p}$ a nonzero prime ideal of $R$, $P\in R[X]$ a monic irreducible polynomial, and $K$ the quotient field of $R$. We give in this paper a lower bound for the $\mathfrak{p}$-adic valuation of the index of $P$ over $R$ in terms of the degrees of the monic irreducible factors of the reduction of $P$ modulo $\mathfrak{p}$. As an important application, when the lower bound is greater than zero for some $\mathfrak{p}$, we conclude that no root of $P$ generates a power integral basis in the field extension of $K$ defined by $P$.