Let be a subfield of . Let be an element of and let where denotes the real root of .
Prove thatif is any Galois extension of contained in then .
I have no idea how to approach this problem.
However, we don't even need this. is a root of x^n-a implies that all roots are of the form for some k (but not the converse, i.e. not all things of this form are the roots of x^n-a), since the minimal polynomial of must divide x^n-a. But use my idea before, that only two of the can be real. This implies that can only have at most two conjugates, hence the degree is <= 2