ORDER OF GAUSS PERIODS IN LARGE CHARACTERISTIC

Abstract

Let $p$ be the characteristic of $\mathbb{F}_q$ and let $q$ be a primitive root modulo a prime $r = 2n + 1$. Let $\beta \in \mathbb{F}_{q^{2n}}$ be a primitive $r$th root of unity. We prove that the multiplicative order of the Gauss period $\beta + \beta^{-1}$ is at least $(\log p)^{c\log n}$ for some $c \gt 0$. This improves the bound obtained by Ahmadi, Shparlinski and Voloch when $p$ is very large compared with $n$. We also obtain bounds for "most" $p$.