Gauss-type sums for cube roots of primes

Question

(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, $\sqrt[3]p$?

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.

Answer 1

The Kronecker--Weber theorem does not guarantee what you claim. Indeed, $\mathbb Q(p^{1/3})$ is not Galois over $\mathbb Q$, and its Galois closure is equal to $\mathbb Q(\sqrt{-3},p^{1/3})$, which is an $S_3$ (and hence non-abelian) extension of $\mathbb Q$. It is an abelian extension of $\mathbb Q(\sqrt{-3})$, but not all extensions of this field are cyclotomic; rather, they are described by the theory of complex multiplication (Kronecker's Jugendtraum).