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Let $L,F,K$ fields such that $L|K$ and $F|K$ are finite extensions, with $F|K$ a Galois extension.

I have to prove that $\text{Gal}(FL|L) \simeq \text{Gal}(F|F\cap L)$

I could prove that $\psi$ such that $\psi(\sigma) = \sigma|_{F}$ is an injective group map between $\text{Gal}(FL|L)$ and $\text{Gal}(F|F\cap L)$

But I don't know how I can prove $\psi$ is surjective

Can someone help me?

ZAF
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  • You could extend a morphism of $Gal(F/F\cap L)$ to a morphism of $Gal(FL/L)$ ? – Fabrizio Oct 19 '20 at 00:35
  • Yes, I tried to do this, but I had trouble verifying the good definition of the extension. $\sigma \in \text{Gal}(F|F\cap L)$, I extend $\sigma$ to $\overline{\sigma}$ such that $\overline{\sigma}(fl) = \sigma(f)l$, I cannot prove that it is well defined – ZAF Oct 19 '20 at 00:39
  • Im sorry, are finite extension, not finite fields – ZAF Oct 19 '20 at 05:44
  • Define your morphism on a basis that do nothing on the elements of L – Fabrizio Oct 19 '20 at 08:09

1 Answers1

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HINT:

Take a Galois extension $E$ of $K$ containing the fields $F$, $L$. Let $G= G(E/K)$. Then $F$, $L$ correspond to $M$, $N$, $FL$ to $M\cap N$, and $F\cap L$ to $\langle M,N\rangle$. Since $F/K$ is Galois, $M$ is a normal subgroup of $G$, so $\langle M,N\rangle=MN=NM$. Now, just use that we have the isomorphism of groups $$N/(M\cap N) \simeq NM/M$$

orangeskid
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