Q(a^1/n)

Let K = Q( a^1/n), where a \in Q and suppose [K: Q] = n. Let E be any
subfield of K and let
[E: Q] = d.Prove that E = Q( a^1/d ).

Couldyou please help me to figure out with this problem.

What i've done : denote alpha = a^1/n. Then consider the norm b=
N_{K/E} (alpha) = alpha^{n/d}* product of n'th roots of unity =
alpha^{n/d} * ksii

Then b in E and b^d = a * ksi^d. Now i don't know how to proceed, i'm
not sure whether ksi is actually a d'th root of unity, it it were, then
we could find a solution of x^d - a in E and that would imply that
Q(a^1/d) in E.

Thanks.

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