I have seen this problem asked by another user but it isn't completely solved in the answers. I'm trying to do it, but I can't.
Question: Suppose $[L:K]=4$ and $charK≠2$ and $L$ is algebraically closed. Show that there is an intermediate field M such that $[L:M]=2$ and that $X^2+1$ splits over $M$. Show that this leads to a contradiction.
I can't show that this M exists, and for this reason, I can't follow with the other parts.
Thanks in advanced.