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, $m \geq 1$, the sum $\sum_{j=1}a_j$ is divisible by $p$. If $d=2p^m$, $p\geq 3$ and $\sum_{j=1}^d|a_d| < p$, then $\sum_{j=1}a_j$ is an even number.