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Let $L$ be a CM field of degree $4$ over the rationals, and let $p$ be a prime number. If $q$ is a power of $p$, I would like to know if it is possible to characterize (in some way) all Weil ${\bf F}_q$-numbers inside $L$.

I was inspired by the corresponding question when $L$ has absolute degree $2$ (i.e., it is an imaginary quadratic field), which has a simple answer: Weil $q$-numbers $\pi$, up to roots of unity in $L$, correspond to principal ideals of norm $q$. The answer in this case is especially simple to get because $L$ has only one archimedean place.

How harder is the problem when $L$ is quartic? Thanks.

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A colleague has told me that the book (in preparation) CM Liftings, by Chai-Conrad-Oort has the answer to the question above in its appendix A.4.8.4 (the book is available after googleing). –  Tommaso Centeleghe Nov 2 '12 at 15:33

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