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# Splitting fields watch

1. Hi,

Any suggestions for the best way to find the degree of the splitting field of:

x^4 + 2x^2 + 2

over Q?

I think I'm happy enough doing easier ones like this but I just can't do these 'more complicated' ones.

My idea so far has been to split the polynomial up into:

(x^2 - ax + sqrt(2))(x^2 + ax + sqrt(2))
where a = sqrt(2 - sqrt(2))

Extending Q to add a is an extension of degree 4, and these polynomials are both in Q(a)... but this is where I get lost. I think this implies it's an extension of degree 8 (since the factors are of degree 2)?

Thanks! Any hints would be greatly appreciated.
2. (Original post by Icy_Mikki)
Hi,

Any suggestions for the best way to find the degree of the splitting field of:

x^4 + 2x^2 + 2

over Q?

I think I'm happy enough doing easier ones like this but I just can't do these 'more complicated' ones.

My idea so far has been to split the polynomial up into:

(x^2 - ax + sqrt(2))(x^2 + ax + sqrt(2))
where a = sqrt(2 - sqrt(2))

Extending Q to add a is an extension of degree 4, and these polynomials are both in Q(a)... but this is where I get lost. I think this implies it's an extension of degree 8 (since the factors are of degree 2)?

Thanks! Any hints would be greatly appreciated.
I'm not sure about the problem itself, but:

is not the original polynomial...

For what it's worth: you can find the roots of the polynomial in ; perhaps that will help?

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Updated: February 8, 2010
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