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.
Categories: math.NT math.AG