Hints for finding a Galois closure

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In summary, the conversation discusses finding the Galois closure of the field \mathbb{Q}(\alpha) over \mathbb{Q}, where \alpha = \sqrt{1 + \sqrt{2}}. The participants determine that the minimal polynomial for \alpha is x^{4} - 2x^{2} - 1, which is separable, therefore a Galois closure exists. They also discuss the approach of finding the splitting field of the minimal polynomial, which will be enough to generate the Galois closure. It is determined that the splitting field is the minimal field that contains all the roots of the minimal polynomial, including the conjugates of \alpha. Therefore, the Galois closure will contain the splitting field
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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.
 
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  • #2
Yes, finding the splitting field will be enough. You're just going to throw in the conjugates of [itex]\alpha[/itex], and that will make your extension normal. Can you guess what they are? One of them is obviously [itex]-\alpha[/itex]. What about the other 2? Considering [itex]\alpha^2[/itex] will be helpful.
 
  • #3
I believe the splitting field is just the original extension adjoin i. So that's handy. But it seems like that really shouldn't be the Galois closure. Why would any Galois extension of the rationals that contains [itex]\mathbb{Q}(\alpha)[/itex] have to contain i? It seems like it would only have to be a subset of the reals.
 
  • #4
There's no reason it should be real at all. Think about it: any normal extension of [itex]\mathbb{Q}[/itex] that contains [itex]\alpha[/itex] will contain the conjugates of [itex]\alpha[/itex], so the minimal such extension is the one you get by adjoining these elements to [itex]\mathbb{Q}(\alpha)[/itex]. And there's nothing saying [itex]\alpha[/itex] can't have non-real conjugates, just because it itself is real.
 
  • #5
I think I understand... The splitting field is the minimal field that contains all the roots of the minimal polynomial, and anything that's Galois over the rationals contains all the conjugates of [itex]\alpha[/itex] (i.e., the roots of the minimal polynomial). So it contains the splitting field. Thanks!

Edit: Er...Galois over the rationals and contains alpha. Heh.
 

What is a Galois closure and why is it important?

A Galois closure is a field extension that contains all the algebraic numbers needed to solve a given polynomial equation. It is important because it allows us to find exact solutions to polynomial equations that may otherwise have only approximate solutions.

How do I know if I need to find a Galois closure for a given polynomial equation?

If the polynomial equation has rational coefficients and is irreducible, then it is likely that a Galois closure is needed to find exact solutions. Additionally, if the polynomial has a degree higher than 4, it is highly likely that a Galois closure is necessary.

What are some tips for finding a Galois closure?

One tip is to start by factoring the polynomial equation over the rational numbers. This can help identify any potential Galois extensions needed. Additionally, using Galois theory and understanding the properties of Galois extensions can also assist in finding a Galois closure.

Can I use computer software to find a Galois closure?

Yes, there are various computer programs, such as Mathematica and Sage, that can calculate Galois closures for polynomial equations. However, it is important to have a basic understanding of Galois theory and field extensions in order to interpret and use the results correctly.

Is finding a Galois closure always necessary?

No, it is not always necessary to find a Galois closure. In some cases, it may be possible to find exact solutions to polynomial equations without using Galois closures. However, for certain equations, it may be the only way to find exact solutions.

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