# Is the subextension of a cyclotomic extension Galois?

If $$\alpha$$ is a root of the polynomial $$f(X)=X^p-a$$ over field $$F$$ of characteristic not $$p$$, where $$p$$ is a prime and $$a$$ is not a $$p$$-th power. Assume that $$F(\alpha) \subset F(\zeta_p)$$, do we have $$F(\alpha)/F$$ is Galois?

• Assuming you're working over a separable field: cyclotomic extensions are abelian... – DonAntonio Jul 4 at 18:05

@DonAntonio essentially answered this in his comment. The extension $$F(\zeta_p)/F$$ is Galois with abelian Galois group, hence every intermediate subfield $$E$$ (i.e. field $$E$$ with $$F\subset E \subset F(\zeta_p)$$) is Galois over F by the Fundamental Theorem of Galois Theory. Here we've used the fact that every subgroup of an abelian group is normal.