I am doing this exercise in Milne's notes and I think it is quite a general stuff so the result would be very useful. But the proof is omitted and the author only left with a sentence:
This is really a group theory problem disguised as a field theory problem.
So may I please ask what is the core of the problem? I had tried to convert it to a group theoric problem but so far I have not solved it. May I please ask for an answer. Or any reference would also be appreciated. Thanks in advance!