Linear relations among roots of a polynomial

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.