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Let $K=\Bbb{Q}(\sqrt{D})$($D$ is a square free negative integer) be a quadratic number field. Class number (order of ideal class group $Cl_K$ of $K$) is $1$ if only if $D=-2,-3,-7,-11,-19,-43,-67,-163$.

I want to know whether the class number $2$ case is known or open. My question is about $Cl_K[2]=\{a\in Cl_K\mid 2a=0\}$.

Are there infinitely many $D$ such that $Cl_K[2]\cong \Bbb{Z}/2\Bbb{Z}$ ?

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    $\begingroup$ Imaginary quadratic fields with small class number have been fully enumerated, see references on Wikipedia. $\endgroup$
    – Wojowu
    Commented Jul 11, 2023 at 9:53

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Are there infinitely many $D$ such that $Cl_K[2] \cong \mathbb{Z} / 2\mathbb{Z}$?

It's well-known (and straighforward to show) that $Cl_K[2]$ has order $2^{r-1}$ where $r$ is the number of prime factors of the discriminant. Since it's clear that there are infinitely many fundamental discriminants with precisely two prime factors, the answer is "yes".

(EDIT: I added a quotation to clarify what exactly I am answering, since the original post hints at several distinct questions with rather different answers.)

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    $\begingroup$ @WillJagy I've added a clarification of exactly what I'm answering (since the title of the question, the first paragraph of the question, and the final paragraph are actually asking three distinct questions) $\endgroup$
    – David Loeffler
    Commented Jul 11, 2023 at 21:41
  • $\begingroup$ Thank you very much. I searched a lot, but I couldn't find any reference for the fact that $\# Cl_K[2]=2^{r-1}$. Could you please provide me with a reference for that? $\endgroup$
    – Duality
    Commented Jul 16, 2023 at 14:13
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    $\begingroup$ It's come up before on this site, see mathoverflow.net/questions/431499/… $\endgroup$
    – David Loeffler
    Commented Jul 16, 2023 at 15:14
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    $\begingroup$ Yes, that's an obvious and well known corollary. $\endgroup$
    – David Loeffler
    Commented Jul 18, 2023 at 9:20
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    $\begingroup$ Yes, this is genus theory. $\endgroup$
    – David Loeffler
    Commented Oct 6 at 8:09

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