I have an arbitrary CM field $K$ and an abelian extension $L$ of $K$ with dimension $[L:K]=p$, with $p$ prim. Suppose there exist a non trivial primitiv root of unity $\xi$ in K.
Must $\xi$ be a $p$-root of unity?
We can first suppose that $\xi$ is a $c$-primitiv root of unity. The element $\xi$ is in $K$, so not really so much to do with the extension. I tried to use the algebraic norm of $L$ over $K$, then $N(\xi)=\xi^p$ because $\xi$ is in $K$. $N(\xi)$ is also a $c$-root of unity.
We can tried some elementary operation of the powers, but I failed to get some interesting result. I don't know how I can use the fact that the extension has dimension $p$..
I'm also not sure that it's work, so if you have a good counter example..
I only need a tipp! Thank you!