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Let $K$ be a number field. If $d$ be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$ then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(K)$ is Hilbert class field $K.$

I understand that it is not true in general. Can I see this working with some assumptions?

Ps. thanks to the comment of Franz Lemmermeyer.

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    $\begingroup$ ${\mathbb Q}(\sqrt{-5})$. $\endgroup$
    – Franz Lemmermeyer
    Commented Jan 14, 2021 at 6:51
  • $\begingroup$ I assume that $d$ should be the largest even integer $\ldots$ $\endgroup$
    – Franz Lemmermeyer
    Commented Mar 12, 2021 at 12:53

1 Answer 1

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The key word is genus theory. The genus class field of a number field $K$ is the maximal unramified extension $KF/K$ where $F/{\mathbb Q}$ is abelian. Thus number fields with trivial genus class field have the desired property.

Genus theory is quite explicit for cyclic extensions $K/{\mathbb Q}$, so in this case you will get conditions that can be verified easily. For general number fields $K$, not a whole lot is known.

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