Elements of high order in Artin-Schreier extensions of finite fields ${\mathbb{F}}_q$

In this article, we find a lower bound for the order of the coset x+b in the Artin-Schreier extension ${\mathbb{F}}_q[x]\mathrm{/}(x^p-x-a)$, where $b\in {\mathbb{F}}_q$ that satisfies a generic special condition.