Determine the minimal polynomial of $\alpha$ over $\Bbb Q(\sqrt{-2})$ Let $\alpha$ be a root of the polynomial $x^4-4x^2+2.$ 
How should I determine the minimal polynomial of $\alpha$ over $\Bbb Q(\sqrt{-2})$?

Let $\alpha$ be a root of the polynomial $x^4-4x^2+2$
$$x^4-4x^2+2$$
$$\left(x^2 - \sqrt{2} -2\right) \left( x^2 + \sqrt{2}  -2\right)$$
$$\left(\sqrt{2 - \sqrt2} - x\right) \left(\sqrt{2 + \sqrt2} - x\right) \left(\sqrt{2 - \sqrt2} + x\right) \left(\sqrt{2 + \sqrt2} + x\right)$$
Then
$$\alpha = \left(\sqrt{2 \pm \sqrt2} \pm x\right) $$
Where the polynomial $\alpha$ has no solutions in $\Bbb{Q}(i\sqrt2)$
