Splitting fields of finite groups

We give a simpler proof of the Goldschmidt-Isaacs theorem in the case p>2 and find new sufficient conditions for the applicability of the theorem in the case p=2. We thus obtain a theorem giving an estimate for the Schur index of an arbitrary irreducible complex representation of a finite group over the field of rational numbers. The proof of this theorem shows that in practical applications there is no need to verify sufficient conditions for the applicability of the Goldschmidt-Isaacs theorem in the case p=2: they can automatically be assumed to hold. We also prove a theorem on the connection between the realizability of any complex representation over the field of rational numbers of a finite group of odd order of a special type and the possibility of constructing regular polygons with straightedge and compasses.