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I know if I have a polynomial $f(x) = g(x) \cdot h(x) \cdot k(x)$ and $g(x),h(x),k(x)$ are polynomials of degree 2, then the Galois group will be $Z/2Z \times Z/2Z \times Z/2Z$ if the roots of $g,h,k$ are distinct.

But what happens in the case when you have mixed factors... i.e. for example the polynomial

$(x^2 + 1)(x^2 -4x + 7)(x^2 - 2)$

the splitting field would be $Q(\sqrt{2},\sqrt{3},i)$ but what would be the Galois group?

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    Did there use to be a question here?2011-11-03
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    This is kind of confusing, someone put the question mark somewhere back in there please?2011-11-03

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