Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
1  
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

1 Answer 1

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 $p$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.