Cyclotomic function fields over finite fields with irreducible quadratic modulus

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 $F({\mathrm{\Lambda }}_M)$ 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 $F({\mathrm{\Lambda }}_M)$ in odd characteristic, extending results of \cite{MXY2016} where the automorphism group of $F({\mathrm{\Lambda }}_M)$ over ${\mathbb{F}}_q$ was computed.