On real quadratic function fields of Chowla type with ideal class number one

Let $\mathbb{F}_q$ be the finite field with $q$ elements, (2${\not|}q$), $k= \mathbb{F}_q(x)$, $K=k(\sqrt{D})$ where $D=D(x) =A(x)^2+a$ is a square-free polynomial in $\mathbb{F}_q[x]$ with $\deg A(x)\geq 1$ and $a\in \mathbb{ F}_q^*$. In this paper several equivalent conditions for the ideal class number $h(O_K)$ to be one are presented and all such quadratic function fields with $h(O_K)=1$ are determined.