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I need help figuring out how to start on the following problem.

Let $k \subset K$ be a finite Galois extension with Galois group $G(K/k)$, let $L$ be a field with $k \subset L \subset K$, and set $H = \{\sigma \in G(K/k)\mid \sigma(L) = L\}$.

(a) Show that $H$ is the normalizer of $G(K/L)$ in $G(K/k)$.

(b) Describe the group $H/G(K/L)$ as an automorphism group.

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Let $N$ be the normalizer of $G(K:L)$ in $G(K:k)$.

$H\subset N$

$h\in H, g\in G(K:L), x\in L, hgh^{-1}(x)=h(gh^{-1}(x))$, since $h(L)=L, h^{-1}(x)\in L$ and $g(h^{-1}(x))=h^{-1}(x)$ since $g\in Gal(K:L)$ we deduce that $hgh^{-1}(x)=h(h^{-1}(x))=x$.

$N\subset H$

Let $n\in N, g\in Gal(K:L), x\in L$,$g(n(x))=nn^{-1}gn(x)$ since $N$ is the normalizer of $Gal(K:L), n^{-1}gn\in Gal(K:L), ngn^{-1}(x)=x$ and $gn(x)=nn^{-1}gn(x)=n(x)$, hus $n(x)$ is in the subfield fixed by $Gal(K:L)$ which is $L$, by the Galois fundamental theorem.

Consider the map $\phi:H\rightarrow Gal(L:k)$ such that $\phi(h)$ is the restriction of $h$ to $L$, the kernel of $\phi$ is $Gal(K:L)$ and $\phi$ is surjective since the extension $K:L$ is separable.

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  • $\begingroup$ What do you mean by the restriction of $h$ to $L$? $\endgroup$ – user346096 Aug 2 '16 at 14:57
  • $\begingroup$ The map $h':L\rightarrow L$ defined by $h'(x)=h(x)$. $\endgroup$ – Tsemo Aristide Aug 2 '16 at 14:59

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