# basis and dimension of the splitting field of $x^4+5x^2+6$

Please help me finding the basis and dimension of the splitting field of the polynomial $x^4+5x^2+6$ in the rational field.

Thanks

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Step one: factor that polynomial. I'm sure you can do that. – Gerry Myerson Feb 6 '13 at 5:52

Hint: You can factor this as $(x^2+3)(x^2+2)$ so the splitting field is $\mathbb{Q}(\sqrt{-2},\sqrt{-3})$. You should then apply the tower rule to get that $[\mathbb{Q}(\sqrt{-2},\sqrt{-3}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{-2},\sqrt{-3}][\mathbb{Q}(\sqrt{-2}):\mathbb{Q}]$. To find each of the degrees in the product try and find the degree of the minimal polynomial. The hint for this is that a quadratic annihilates things and so you'r each degree 1 or 2 at each step--then recall that you're degree one only if you were in the field all along!