It's an exercise from "Advanced Modern Algebra" Rotman. Exercise 8.44 on page 574.
The problem is, prove that the degrees of the irreducible representations of $G$ over $K_1$ are the same degrees of the irreducible representations of $G$ over $K_2$, where $G$ is a finite group, and $K_1$ and $K_2$ are algebraically closed fields, characteristics char$K_1$=$p$ and Char$K_2$=q do not divide $\lvert G\rvert$.
I have known that $K_iG$ is semisimple from Maschke's Theorem, and that the number of simple components are the same (equal to the number of conjugacy classes of $G$). But I don't know how it can be proved that the decompositions are actually equivalent.
Thank you for helping!