[Some (useless) context: the following problem comes from a problem in Algebraic Geometry, where I have to show that a certain morphism $\textbf P^2\to \textbf A^2$ is inseparable of degree $2$.]
Let $s,t,x,y$ be indeterminates and let $k$ be a field (algebraically closed if it helps) of char $2$. I define a monomorphism of fields $j:k(s,t)\to k(x,y)=:L$ by sending
\begin{equation} s\mapsto j(s)=\frac{x^2+xy}{y+1} \\ t\mapsto j(t)=\frac{x^2+x}{y(y+1)}. \end{equation}
Let us call $K$ the image of $j$, i.e. $K:=k(j(s),j(t))\subset L$. I would like to show that $L/K$ is inseparable of degree $2$.
To do so I just observed that $x\in K(y)$, so it remains to find the minimal polynomial of $y$ and check it is inseparable of degree $2$. Does anyone have a hint? I tried to write $y^2=\alpha y+\beta$ for some $\alpha,\beta\in K$ but I did not succeed (and of course this would not be enough).
Thanks in advance.