Totally \(p\)-adic algebraic numbers of degree 4 (Q6170580)

Let~\(p\) be a prime number~\(p\). An algebraic number~\(\alpha\) is said to be totally \(p\)-adic if all the roots of its minimal polynomial lie in~\(\mathbb Q_p\). Denote by~\(\tau_{n,p}\) the minimal logarithmic Weil height of a totally \(p\)-adic number of degree~\(n\) which is not a root of unity. \textit{E. Stacy} proved in [Open Book Ser. 4, 387--401 (2020; Zbl 1472.11325)] that if \(p>3\), one has that \(\tau_{n,p}\leq 0.703762\). This article establishes that \[ \tau_{4,p}\leq \frac{\log(5)}{4}=0.40236. \]