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Let $K/F$ be a Galois extension of number fields. Is it true that the Hilbert class field $H_K$ of $K$ is an extension of the Hilbert class field $H_F$ of $F$ ?

If the class number of $F$ is $h_F > [K:F] h_K$, then we find a counterexample, but I don't know how to check this.

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Yes, since the compositum $K\cdot H_F$ is an unramified Abelian extension of $K$, and so is contained in $H_K$.

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  • $\begingroup$ Thank you. Why is $KH_F / K$ unramified? I only see that if $v$ is a place of $K$ that ramifies, then $v\vert_F$ is a place of $F$ ramifying already in $K$. $\endgroup$ – Alphonse Sep 7 '18 at 16:15
  • $\begingroup$ @Alphonse Just localise the whole setup at your place $v\mid_F$ and then all should be clear. $\endgroup$ – Lord Shark the Unknown Sep 7 '18 at 16:20
  • $\begingroup$ Apparently, $KH_F/K$ is unramified by Prop 173 here. $\endgroup$ – Alphonse Sep 7 '18 at 16:38
  • $\begingroup$ Two more questions : 1) The fact $H_F \subset H_K$ holds even if $K/F$ is not Galois, right? 2) So in general, we have $h_F/h_K \leq [K:F]$, right? Many thanks! $\endgroup$ – Alphonse Sep 7 '18 at 16:38
  • $\begingroup$ May I have some hints for my 2 little questions above, please? Then I think that I can accept your answer. $\endgroup$ – Alphonse Sep 8 '18 at 7:51

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