fundamental rule of galois theory

I don't know if the proof is right because answers to Rotman's exercises are hard to find (for me)

Tbh this should be marked as NSFW, math gore

EDIT: re-read it and I didn't mention that there are no factors that permute roots between g and h since the intersection of their Galois groups is trivial, also not sure if xi and gamma are even elements in the subgroup of S_n (they should be cause the proposition is true, idk how to explicitly show D:)