Why is a Galois extension over ??
Intersection of subfields (in a larger field) is a subfield. Therefore, is a subfield (of the algebraic numbers, for example). Now is a cyclotomic extension and therefore is an abelian group. Since all subgroups of this group are normal it follows that if then is a Galois extension by the fundamental theorem. Since the rest follows.
in general if is a family of (finite) Galois extensions of then is also Galois over just recall that a (finite) extension is Galois iff it's separable and normal.
Edit: what was i thinking?!! lol ... this, although true, but doesn't apply to Stiger's problem! thanks ThePerfectHacker for pointing that out.