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Could someone help me with the following question? Let $p$ be a prime number and consider

$x=p^{\frac{1}{p-1}}$

Does $x$ belong to the cyclotomic field $\mathbb{Q}(\mu_p)$?

Thanks a lot!

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    x generates an extension that is clearly not normal when p > 3.2012-11-09
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    Although the answer to your question is "no" when $p$ is odd, over the $p$-adic numbers there is something close to this: for all primes $p$, the number $(-p)^{1/(p-1)}$ lies in ${\mathbf Q}_p(\mu_p)$. That is, $X^{p-1} + p$ has a root (and in fact a full set of roots) in ${\mathbf Q}_p(\mu_p)$.2012-11-10

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