If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))?

Clearly we need to have distinct roots of unity otherwise this won't work!

Correction: Let assume that Z is not rational itself otherwise obviously it's wrong. Here I hope to extend the proof of Kronecker thm! I have "Z is a sum of t distinct roots of unity and |Z| is a rational integer" I conjecture that either Z is rational or a root of unity!