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!
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!
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 $p$.