Lineare Abhängigkeiten von Einheitswurzeln. (Linear dependence of roots of unity.) (Q1062095)

The author establishes the dimension of the \({\mathbb{Q}}\)-vector space of the roots of a cyclotomic polynomial. He remarks that this problem originates in the notice 40 from \textit{C. F. Gauss}' ''Mathematisches Tagebuch'' (1976; Zbl 0327.01009) (9. October 1796). It is proved that the m-primitive roots of unity are linearly independent if and only if m is square-free. This is applied to obtain a result connected with some theorems of L. Kronecker, H. Weber and D. Hilbert.