Why is a Galois extension over ??

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- Dec 21st 2008, 02:23 AMStigerGalois extension over Q
Why is a Galois extension over ??

- Dec 21st 2008, 08:46 AMThePerfectHacker
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.

- Dec 21st 2008, 01:41 PMNonCommAlg
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. - Dec 21st 2008, 04:08 PMThePerfectHacker