Question put in mathstackexchange but received no answer.

It is well-know( see Goldschmidt book: Algebraic Functions and Projective Curves) that a for $q$ a power of $2$ a quadratic separable extension of $\mathbb F_q(T)$ can be written as: $$K:=\mathbb F_q(T)[y]\text{ with }y^2+y=f(T)\text{ and } f(T)\in\mathbb F_q(T).$$ Moreover $f(T)=\frac{R(T)}{P_1(T)^{2d_1-1}\cdots P_s(T)^{2d_s-1}}$ with $P_i$ irreducibles of $\mathbb F_q[T]$, $d_i$ positive integers, and $R(T)\in\mathbb F_q[T]$ such for all $i$ $P_i\nmid R$. I am looking for describing the integral closure $\mathcal O_K$ of $\mathbb F_q[T]$ in $K$.

Let $x=a+by\in\mathcal O_K$ ($a,b\in\mathbb F_q(T)$). One has $\mathrm{Tr}(x)=b\in\mathbb F_q[T]$ and $N(x)=(a+by)(a+b(y+1))\in\mathbb F_q[T]$. Hence, $a^2+b^2f(T)+ab\in\mathbb F_q[T]$

I am stuck here. I did not manage to obtain conditions on $a$. Thanks in advance for any hint or any solution that you could give me.