I've been thinking about this question and I know that if $L:K$ is normal and finite where L, K are subfields of $\mathbb{C}$, then $|\Gamma(L:K)| = [L:K]$. I was wondering if, assuming that L:K is a finite extension such that $|\Gamma(L:K)| = [L:K]$, can you conclude L:K necessarily normal?
I don't know how I would even start this / if there is an easy counterexample if it is false so any help would be much appreciated! Thank you.