# Behaviour of the least root modulo a prime of a polynomial

## TL;DR

This paper proposes a conjecture about the distribution of the smallest root modulo primes for polynomials with roots that are algebraically independent, focusing on primes where the polynomial fully splits.

## Contribution

It introduces a new conjecture on the distribution of the least root modulo primes for certain polynomials, expanding understanding of polynomial root behavior over finite fields.

## Key findings

- Conjecture on the distribution of the least root modulo primes.
- Analysis of polynomials without linear relations among roots.
- Focus on primes where the polynomial fully splits.

## Abstract

For a polynomial $f(x)\in\mathbb Z[x]$ without non-trivial linear relations among roots, we propose a conjecture on the distribution of the least root $r_p$ ($r_p\in\mathbb Z,\,0\le r_p<p)$ of $f(x)\equiv0\bmod p$ where $p$ runs over the set of primes such that $f(x)$ modulo $p$ is fully splitting.