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## Homework Statement

Find the Galois closure of the field [itex]\mathbb{Q}(\alpha)[/itex] over [itex]\mathbb{Q}[/itex], where [tex]\alpha = \sqrt{1 + \sqrt{2}}[/tex].

## Homework Equations

Um...the Galois closure of E over F, where E is a finite separable extension is a Galois extension of F containing E which is minimal.

## The Attempt at a Solution

I've found that the minimal polynomial of [itex]\alpha[/itex] is [itex]x^{4} - 2x^{2} - 1[/itex] (at least I think. I'm having a heck of a time showing it's irreducible over the rationals but I'm willing to just leave that as an exercise for the reader, so to speak), and it's separable, so I know a Galois closure exists. But I can't figure out a good way to generate it.

There are no examples of actually doing this explicitly in the book, though the proof of existence implies that you should take the intersection of all Galois extensions of F containing E. Or would finding explicitly the splitting field of the minimal polynomial be enough in this case? If so, could you guys help me out with that? This is kind of tricky.