Let be the Galois closure of , where and suppose .
Prove that or
( Euler's function)
i haven't thought about the details yet but what is clear to me is that the Galois closure of would be the splitting field of the polynoimal because which means we've
assumed that is irreducible over so if we let then thus so what you need to prove
is that the only possible values of are and
Yes, but show do you prove this?
Notice by the Natural Irrationalities theorem: Let and and so .
Therefore, .
But, is a subfield of both and . But subfields of have form , . Thus, somehow the only possibilities for this degree are . But there does not seem to be direct way to show this. This there some kind of theorem about real subfields of a cyclotomic extension?
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The other thing I see is that is an -Kummer extension. Therefore, the degree is equal to the order of the subgroup in where . But this seems to be really messy.