(Edit: now crossposted on MathOverflow: https://mathoverflow.net/questions/278869/problem-with-a-proof-of-wilsons-profinite-groups)
I need help with the proof of Proposition (3.1.3) given by Wilson in his 'Profinite Groups', which states the following:
Let $M_1, M_2$ be intermediate fields of a Galois extension $K/F$ and let $\gamma : M_1 \to M_2$ be a field isomorphism fixing each element of $F$. Then $\gamma$ can be extended to an automorphism of $K$.
Here's the proof given in the book:
$\textit{Proof}$. For each $N \in \mathcal F := \left\{L \mid F \leq L \leq K,\: L/F \textrm{ finite Galois extension}\right\}$, define $$B_N := \left\{{\left(\sigma_L\right)}_{L \in \mathcal F} \in \prod_{L \in \mathcal F} \textrm{Gal}\left(L/F\right) \:\mid {\sigma_N}_{|N \cap M_1} = \gamma_{|N \cap M_1},\:\forall L \leq N :\, \sigma_L = {\sigma_N}_{|L}\right\}.$$ Then every $B_N$ is non-empty, closed in $\prod_{L \in \mathcal F} \textrm{Gal}\left(L/F\right)$ and no finite intersection of such sets is empty. Thus $B := \bigcap_{N \in \mathcal F} B_N$ is non-empty and it is subset of $\varprojlim_{L \in \mathcal F} \textrm{Gal}\left(L/F\right)$. $[\ldots]$
The conclusion of the proof sounds then clear, as it does the fact that every $B_N$ is non-empty and (once proved the preceding statements) that $B$ is non-empty, since every $\textrm{Gal}\left(L/F\right)$ is finite and discrete, hence compact. Instead, I cannot completely figure out why each $B_N$ is closed: I thought we could exploit the fact that if $f, g : X \to Y$ are continuous maps from a topological space $X$ to a Hausdorff space $Y$, then $\left\{x \in X \mid f(x) = g(x)\right\}$ is closed in $X$ (and the fact that intersection of closed is closed), but I'm not completely sure it works: it seems that the extension $(N \cap M_1)/F$ need not be necessarily normal, and so if we take $\sigma \in \textrm{Gal}\left(N/F\right)$ we are not guaranteed (at least, by results I am aware of) that $\sigma_{|N \cap M_1} \in \textrm{Gal}\left((N \cap M_1)/F\right)$, and so cannot talk about continuity of $\sigma \mapsto \sigma_{|N \cap M_1}$.
Any detailed suggestions about how to prove that each $B_N$ is closed (and, probably, also that any finite intersection of them is not empty and that $B$ is a subset of the inverse limit) or, in alternative, any other proofs of the given Proposition would be highly appreciated (do note that Galois correspondence is assumed only for finite Galois extensions, this result being then used in the book to generalize it also to infinite ones). Thanks in advance!