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Let $x\in\{\text{totally ramified, inert, totally split}\}.$

If $p\geq 5$ is a prime, are there infinitely many imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of class number coprime to $p$ so that $p$ has ramification behaviour $x$ in $K/\mathbb{Q}$?

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    $\begingroup$ Should be "yes" for all three cases, but I cannot find an argument for the "class number prime to $p$" condition. $\endgroup$
    – WhatsUp
    Commented Aug 1, 2016 at 11:22
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    $\begingroup$ Is it known that there are infinitely many imaginary quadratic fields with class number coprime to, say, 17? $\endgroup$
    – Gerry Myerson
    Commented Aug 1, 2016 at 13:36

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A more general version of this statement was shown by Kimura in Acta Arith. (2003). His corollary gives $\gg \sqrt{X}/\log X$ such fields ${\Bbb Q}(\sqrt{-d})$ with $d\le X$, and also allows you to add further splitting conditions. There is an extensive literature on divisibility and indivisibility of class numbers and Kimura's paper has many other relevant references.

Update There is also an erratum to Kimura's paper -- it seems that he needs the existence of one such field to get a lower bound for the number of such fields; for large $p$ this is guaranteed by work of Horie. In the meantime a paper of Wiles has addressed the general version of this question, and there is a quantification of Wiles's result due to Beckwith.

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  • $\begingroup$ That is an amazing result indeed! Thank you. $\endgroup$
    – The Thin Whistler
    Commented Aug 1, 2016 at 16:38

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