Linear relations among roots of a polynomial

TL;DR

This paper investigates the linear relations among roots of monic polynomials over rationals, especially in abelian extensions, revealing their structure and implications for root distribution modulo primes.

Contribution

It provides a group-theoretic analysis of linear relations among roots, fully characterizing the structure when the field extension is abelian.

Findings

Complete understanding of linear relations in abelian extensions

Insights into roots distribution modulo primes

Group-theoretic framework for root relations

Abstract

Let $f(x)$ be a monic polynomial over $\mathbb{Q}$ with complex roots $\alpha_1,\dots,\alpha_n$. Linear relations among them and $1$ over $\mathbb{Q}$ play an important role when we study the distribution of roots modulo a prime. We study linear relations group-theoretically. If $\mathbb{Q}(\alpha_1)$ is an abelian extension, we can understand the structure completely.