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I have used sage to compute the class number of the number field generated by the polynomial $x^n-2$ for small $n$. Specifically, setting proof=False, (which hopefully just means that GRH is assumed), the class number is $1$ for all $n\leq 40$. This seems slightly surprising.

I have two questions:

  1. Does anyone know if these class numbers have been computed for larger $n$? If so do we know a $n$ for which it isn't $1$?

  2. Heuristically, how should the class numbers behave? Presumably there should be infinitely many $n$ for which its $1$, but is there a heuristic justification of why I haven't found a single $n$ where its not $1$. Obviously I've only considered very small $n$ but with other families of fields, real quadratic for example, its quite easy to find examples with class number $>1$.

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    $\begingroup$ Interesting question, but your phrasing is ambiguous. Do you mean the field generated by one root of the polynomial $x^n-2$ (as the title suggest) or all roots? I guess it means the former, as the latter contains the corresponding cyclotomic field and is unlikely that, e.g.for $n=23$, the divisor classes will become principal in the extension. $\endgroup$ Mar 7, 2013 at 13:44
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    $\begingroup$ Not directly related, but here's another pattern for the fields ${\mathbf Q}(\sqrt[n]{2})$ that works for all $n \leq 1000$ but eventually has counterexamples: the ring of integers is ${\mathbf Z}[\sqrt[n]{2}]$. The first time this is false is $n = 1093$. $\endgroup$
    – KConrad
    Mar 7, 2013 at 14:01
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    $\begingroup$ A conjecture going back to Weber is that the number fields ${\mathbf Q}(\zeta_{2^n}+\zeta_{2^n}^{-1})$ have class number 1 for all n, although it has only been proved unconditionally for $n \leq 5$. Fukuda and Komatsu shows any prime factor of one of these class numbers is greater than 108. Look at the answers and comments to mathoverflow.net/questions/82480/… $\endgroup$
    – KConrad
    Mar 7, 2013 at 14:08
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    $\begingroup$ See also mathoverflow.net/questions/88288/class-numbers-of-mathbfq21-n $\endgroup$
    – user19475
    Mar 7, 2013 at 15:03
  • $\begingroup$ I do not yet vote to close as a duplicate, waiting for clarfication what precisley is meant here, but if Felipe Voloch's interpretation is right this seems too close to Timo Keller's question to have it in addition.(One might add the numerical data from here, there.) $\endgroup$
    – user9072
    Mar 7, 2013 at 15:45

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