On the exact degree of multi-cyclic extension of $\mathbb{F}_{q}(t)$

Abstract

Let $q$ be a power of a prime number $p$, $k=\mathbb{F}_{q}(t)$ be the rational function field over finite field $\mathbb{F}_{q}$ and $K/k$ be a multi-cyclic extension of prime degree. In this paper we will give an exact formula for the degree of $K$ over $k$ by considering both Kummer and Artin-Schreier cases.

Notice that this paper will not be published. We put it on the arXiv, since we hope that some character sums estimations over $\mathbb{F}_{q}[t]$ in Sections 2 and 3 of this paper may be useful for someone in the future