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:
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 $1$?
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 $>1$.