(Quadratic) Gauss sums express square root of any integer as a sum of roots of unity (or of cosines of rational multiples of $2\pi$, if you will) with rational coefficients.
But Kronecker-Weber guarantees that any root of any integer can be expressed as a sum of that kind.
What is the It there corresponding formula for, say, $\sqrtp$?
Upd. I'm sorry, but original question doesn't make much sense. The question I, perhaps, meant to ask is (as Matt E kindly points out) discussed in David Speyer's answer.