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A classic conjecture of Gauss, which also goes by the name of "Class Number One Problem", asserts that there are infinitely many primes $p \equiv 1 \pmod{4}$ such that the quadratic field $\mathbb{Q}(\sqrt{p})$ has class number equal to one. This is of course still open.

What if we widen the net to include ALL number fields? Do we know of a single infinite family (with the degree allowed to increase to infinity) of number fields where each member of the family has class number one?

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    $\begingroup$ No infinite family has been proved to work, but there are proposed examples of them in $\mathbf Z_p$-extensions: see mathoverflow.net/questions/82480/… including the comments there. $\endgroup$
    – KConrad
    Commented Jun 5 at 17:20
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    $\begingroup$ One can imagine a solution to this problem that is not exactly explicit. By class field theory, the class group of a number field is isomorphic to the Galois group of the maximal unramified abelian extension. One may take this extension, then iterate, taking the maximal unramified abelian extension at each stage. If this process terminates, then the final field has class number one. One can imagine proving finiteness for some class of fields without knowing explicitly the extension. $\endgroup$
    – Ian Agol
    Commented Jun 7 at 22:35

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We don't know that there are infinitely many number fields with Class Number one, so a fortiori we don't know any explicit infinite family of such number fields.

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