A curiosity concerning the degrees of the characters of a finite group

Abstract:

Let G be a finite group with irreducible characters $\{ \ldots ,\chi , \ldots \}$ and $K = {\mathbf {Q}}( \ldots ,\chi , \ldots )$ the field generated over the rationals by their values. We will prove: If $K = \mathbf {Q}$ (or if $[K:{\mathbf {Q}}]$ is odd) then $\prod \limits _{\chi (1)\;odd} {\chi (1)}$ is a perfect square.