# About roots of unity in an arbitrary CM field K with an abelian extension L

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!

• "extension $L$ of $L$" – should that be, "extension $L$ of $K$"? – Gerry Myerson Oct 15 '18 at 7:55
• Yeah sorry, I was not concentrate :) – Oslap Oct 15 '18 at 8:10