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Calculating the class numbers of $\mathbf{Q}(2^{1/n})$ for small $n$ always yields $1$. Is it true for an infinite number of $n$s? Does applying Iwasawa theory to the false Tate curve tower $\mathbf{Q}(2^{1/p^n}, \mu_{p^n})$ help?

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I seem to recall that this was once conjectured by Harvey Cohn. In absence of any numerical support for large n I tend to believe that the conjecture is wrong, however. – Franz Lemmermeyer Feb 12 '12 at 18:07
Since there is no infinite set of number fields in which all are proved to have class number 1, the answer to the first question is "not yet known". And, Timo, exactly what range of $n$ have you checked? – KConrad Feb 13 '12 at 0:16
On a marginally related note, the ring of integers of ${\mathbf Q}(\sqrt[n]{2})$ for $n$ up to 1000 turns out to be ${\mathbf Z}[\sqrt[n]{2}]$, but this pattern eventually breaks down. It is related to finding solutions to $2^{p-1} \equiv 1 \bmod p^2$ when $p$ is prime (Wieferich primes). The first such prime is 1093 and that's the first $n$ for which the integers are more than ${\mathbf Z}[\sqrt[n]{2}]$. – KConrad Feb 13 '12 at 0:21

As KConrad says, the answer to the first question is "not yet known". For Iwasawa theory, you could consider the extension $L_\infty=\mathbb{Q}(\sqrt[p^\infty]{2},\zeta_{p^\infty})$ which is Galois over $\mathbb{Q}$ with Galois group isomorphic to $G=\mathbb{Z}_p^\times\ltimes \mathbb{Z}_p$ and some Iwasawa theory can tell you the structure (may be assuming that $2$ is a multiplicative generator $\pmod{p^2}$) of the $p$-part of the class group of your field, but nothing for $\ell\neq p$. Already in the most basic situation where you consider the cyclotomic $\mathbb{Z}_p$-extension of the rationals, Iwasawa theory tells you that $p$ never divides the class number, and also that for all $\ell\neq p$ the power of $\ell$ dividing the class number of the $n$-th layer is bounded independently of $n$ (but may depend on $\ell$): this is a theorem of Washington. But it is still unknown whether only finitely many such $\ell$ do actually occur.

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