I am reading Stichtenoth's book on function fields and codes and have just arrived at the discussion of divisors and the divisor class groups for function fields. I'm interested in how these are in analogy with ideals and the ideal class groups of number fields.
If $F$ is a function field with constants $\mathbb{F_q}$ then we can define a quadratic extensions as one might in the number field world by taking some square-free $d$ in the integral closure of $\mathbb{F}_q[T]$ in $F$ and forming the quadratic extension $K=F(\sqrt{d})$. Likewise, let $K'=F(\sqrt{d'})$ for some $d\neq d'$.
Question: Are the data of the divisors $(d)$ and $(d')$ in $\text{Div}(F)$, or their classes in $\text{Cl}(F)$, enough to determine if $K \cong K'$ or $K \not\cong K'$? Similarly, is there an analogue for the discriminant but in the language of divisors?
I see that the question could be formulated more generally for extensions of function fields, but would like to understand this (simpler) case first.