Polynomials over Ring of Integers of Global Fields that have Roots Modulo Every Finite Indexed Subgroup

TL;DR

This paper characterizes polynomials over the ring of integers in global fields that have roots modulo every finite-indexed subgroup, providing criteria based on Galois groups and explicit constants.

Contribution

It introduces two new criteria for intersective polynomials over global fields, one involving Galois groups and another based on computable constants.

Findings

Provides criteria for intersective polynomials using Galois group properties

Develops verifiable conditions involving constants depending on the field and polynomial

Uses Chebotarev density theorem to establish bounds on prime ideals

Abstract

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.