No three algebraic conjugates of degree sixteen sum to zero

Abstract

Let $d$ be the smallest positive integer, not a multiple of $3$, for which there exists an algebraic number $\al$ of degree $d$ over $\mathbb{Q}$ whose three algebraic conjugates add to zero. We prove that $d=20$. This is derived from the following result: for any linear relation $\sum_{j=1}^d a_j \al_j=0$ with coefficients $a_j\in \mathbb{Z}$ among the conjugates $\al_j$ of an algebraic number of degree $d=p^m$, where $p$ is a prime number, $m\ge 1$, the sum ${\sum }_{j=1}{a}_j$ is divisible by $p$. If $d=2p^m$, $p\ge 3$ and ${\sum }_{j=1}^d|a_d|<p$, then ${\sum }_{j=1}{a}_j$ is an even number.