For $char(K)=2$ and some $\alpha \in L$ \ $K, \alpha^2\in K$ show that $L/K$ is quadratic, i.e. $[L:K] = 2$
I know it has something to do with the characteristics being equal to 2, but no idea where to start. Any hints?
EDIT:
The exercise I am trying to solve is:
Quadratic extension $L/K$ is inseparable $\iff$ $char(K)=2$ and $\exists \alpha \in L$ \ $K$ with $\alpha^2 \in K$
I think that I have a good solution for $\Rightarrow$, but for $\Leftarrow$ I manage to show $L/K$ is inseparable by the fact that $f = Irr(\alpha,K) = x^2 - \alpha^2$. There exists then a $g \in K[x], g = x - \alpha^2$. It holds that $f(x) = g(x^2)$ and because $char(K)=2$, it holds that $f$ is inseparable and thus $\alpha$ is inseparable. Which also proves $L/K$ is inseparable. But I don't know how to show it is quadratic, hence my question.