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When I was doing class group computations for number fields $K:\mathbb{Q}$, I realised that the cyclic group comes up a lot. I was thinking about the following question:

Suppose we have a $n$ fixed, is there an algorithm that can be used to find a number field $K$ such that the class group, $Cl(K)$ is isomorphic to $C_n$, the cyclic group of order $n$?

I thought about this but I couldn't really prove this or disprove with a counter argument. Also I saw that there is a generalisation called the Cohen-Lenstra heuristics, however I was wondering since I am only considering a special case of this, would my question becomes easier to solve?

Many thanks in advance!

amWhy
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UnsinkableSam
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    There is a Theorem due to Claborn saying that any abelian group is the class group of some Dedekind domain. Of course, this does not say much about class groups of number fields in particular (those class groups are known to be finite for one). I would be somewhat surprised if the answer to your question is unknown, though. – Jyrki Lahtonen Jun 20 '21 at 20:14
  • I seem to be collecting questions about ANT I would like to know more into the Pearl Dive. It was intended to be more diverse, but the level of activity there has been low lately. – Jyrki Lahtonen Jun 30 '21 at 12:01
  • @JyrkiLahtonen Thank you so much Jyrki, for both trying to solve this problem and attracting more attention to this problem! Much appreciated! – UnsinkableSam Jul 01 '21 at 14:57
  • No problem. Now I'm wondering how much ground can be covered by quadratic imaginary extensions alone? That's a well studied case. Perhaps because rather elementary techniques work there. See 1 and 2. Those techniques should be generalizable. – Jyrki Lahtonen Jul 01 '21 at 15:04

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