# Generic polynomials for cyclic function field extensions over certain finite fields

## TL;DR

This paper determines all generic polynomials for specific cyclic function field extensions over finite fields where the field size satisfies certain congruence conditions, expanding understanding of such algebraic structures.

## Contribution

It explicitly finds all generic polynomials for geometric yclic function field extensions over finite fields with particular size and characteristic constraints.

## Key findings

- Identifies all generic polynomials for the specified cyclic extensions.
- Provides explicit constructions under given finite field conditions.
- Enhances the classification of function field extensions with cyclic Galois groups.

## Abstract

In this paper, we find all the generic polynomials for geometric $\ell$-cyclic function field extensions over the finite fields $\mathbb{F}_q$ where $q= p^n$, $p$ prime integer such that $q \equiv -1 \mod \ell$ and $(\ell , p)=1$.