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.


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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.