Degrees of irreducible representations are the same for all splitting fields

Statement

Suppose $G$ is a finite group. Then, given any splitting field $K$ for $G$ (so the characteristic of $K$ does not divide the order of $G$) it is possible to construct a bijection between the irreducible representations of $G$ over $K$ and the irreducible representations of $G$ over the field of complex numbers, such that the bijection preserves the degree of a representation.

In particular, this means that the Degrees of irreducible representations (?) for a finite group are the same for all splitting fields (characteristic not dividing the group order).