Let $L/K$ be a finite separable extension of a field and let $M$ be its Galois closure (i.e. the minimal degree extension of $L$ for which $M/K$ is Galois). Show that the set of embeddings (injections) $hom_K(L, M)$ of $L$ in $M$ which fix $K$ is in a natural bijection with the set of right cosets of $Γ(M : L)$ in $Γ(M : K)$.
I tried some "natural" bijections (e.g. associate $\varphi$ to $\sigma_{\varphi}$ which restricted on $Im \varphi$ behaves as taking the preimage for $\varphi$), but they can't work as I don't use the minimality of $M$. Also, when I say that $Γ(M : L)\sigma = Γ(M : L)\tau$ implies $\sigma = g\tau$ for some $g\in Γ(M : L)$, does this mean that $\sigma(m) = g(\tau(m))$ for all $m\in M$ or $\sigma(m) = \tau(g(m))$ in $M$?
Any help appreciated!