Let so that . Define the polynomial . Notice that . Since can be factored as into linear factor in by Gauss' Lemma it means it can be factored into linear factors in . Thus, are its only factors and so by the factor theorem it means . Q.E.D.
I guess this shows the following generalization: if are rational and where are the elementary symettric functions for it means .
So in particular if it means each individual one must be an integer.