Linear relations between three algebraic conjugates of degree twice a prime

In this paper, we show that there is no irreducible polynomial \(f(x)\) of degree \(2p\) (\(p\geq5\) is a prime number) over \({\mathbb Q}\) whose three distinct roots sum up to zero. This extends some earlier results on linear relations between three algebraic numbers. In particular, let \(d\) be the smallest positive integer not a multiple of \(3\), for which there exists an irreducible polynomial \(f(x)\) of degree \(d\) whose three distinct roots add up to zero. In 2015, Dubickas and Jankauskas found that \(10\leq d \leq 20\). As a corollary, we show that it is either \(d=16\) or \(d=20\).