# does this number belong to the cyclotomic field?

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!

-
x generates an extension that is clearly not normal when p > 3. – franz lemmermeyer Nov 9 '12 at 14:54
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 ${\mathbf Q}_p(\mu_p)$. – KCd Nov 10 '12 at 2:30

No, this is not true. For example, when $p=3$ the cyclotomic field $\mathbb Q(\mu_3)$ is of degree $\varphi(3) = 2$ over $\mathbb Q$ and $\mathbb Q(\sqrt{3})$ is also of degree 2 over $\mathbb Q$. So if $\sqrt{3}$ were contained in $\mathbb Q(\mu_3)$ these fields would have to be equal. But $\mathbb Q(\sqrt{3})$ is real and $\mathbb Q(\mu_3)$ is not, so this is not the case. I don't think this is true for any odd prime $p$.