Let $K$ be a CM-field, ie. a totally imaginary quadratic extension of a totally real number field $F$ and let $p > 2$ be a rational prime. My question simply is
Are the $p$-th roots of unity, denoted by $\mu_p$ in $K$ or not? (and why?)
My feeling is yes, but I'm not sure about it and have no glue how to prove it. Thank you!
EDIT: As the answer in general is 'no', I'm specifying my question to:
Under what conditions are the $p$-th roots of unity in $K$? (if there are any general conditions), ie. $\Bbb Q(\mu_p) \subset K$?