Cyclotomic function fields over finite fields with irreducible quadratic modulus

TL;DR

This paper characterizes cyclotomic function fields over finite fields with quadratic irreducible moduli and determines their automorphism groups in odd characteristic, extending previous results in the field.

Contribution

It provides a complete characterization of such cyclotomic function fields and explicitly computes their automorphism groups in odd characteristic.

Findings

Characterization of cyclotomic function fields with quadratic irreducible modulus

Full automorphism group determination in odd characteristic

Extension of previous automorphism group results

Abstract

Let $\mathbb{F}_q$ be the finite field of order $q$ and $F=\mathbb{F}_q(x)$ the rational function field. In this paper, we give a characterization of the cyclotomic function fields $F(\Lambda_M)$ with modulus $M$, where $M \in \mathbb{F}_q[T]$ is a monic and irreducible polynomial of degree two. We also provide the full automorphism group of $F(\Lambda_M)$ in odd characteristic, extending results of \cite{MXY2016} where the automorphism group of $F(\Lambda_M)$ over $\mathbb{F}_q$ was computed.