Let $q$ be a power of $2$ and $K$ be the quadratic extension of $\mathbb F_q(T)$ defined by $K:=\mathbb F_q(T)[y]$ where $y^2+y=f(T)\in\mathbb F_q(T)$. Put $\mathcal O_K$ the integral closure of $\mathbb F_q[T]$ in $K$. Can there exist principal prime ideals of $\mathcal O_K$ lying above ramified irreducibles of $\mathbb F_q[T]$?
For now, I did not find any counterexamples, but I have no proof of this claim. Can anyone give an hint or solution of the trueness (or not) of this fact. Thanks in advance
