2 problems

Loxton's conjecture. The quartic Gauss sum satisfies

Let $p\equiv1\pmod8$ be a prime. Let $K=\mathbb{Q}(\zeta_{p-1},\zeta_p)$, let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ above $(1-\zeta_p)\mathbb{Z}[\zeta_p]$, and let…