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 $F(\Lambda_M)$ 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.