On sums of roots of unity

Suppose that a(o) + Sigma(i=1)(k-1) a(i)zeta(ni) = 0 for some root of unity zeta of order Q with (Q, n(1),..., n(k-1)) = 1 and all coefficients a(i) b elonging to a number field L. We bound Q in terms of k and d = [(L boolean AND Q(zeta)) : Q]. This generalizes a result of Conway and Jones for the ca se of rational coefficients. Moreover, we give an application to linear rel ations among characteristic-functions of arithmetical progressions.