Find the splitting Field of $x^4+x^2+1$

Question

Find the splitting field of $$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$ I have $(-1±\sqrt{-3})/2$ and $(1±\sqrt{-3})/2$ so, $\mathbb{Q}(1,\sqrt{-3})$, but i do not make sure about that.

Answer 1

Hint: If $x^2-x+1=0$ then $(-x)^2 +(-x)+1=0$, and conversely.

Answer 2

Your polynomial has roots $\frac{-1+\sqrt{3}}{2}$, $\frac{-1-\sqrt{3}}{2}$ as well as two complex cube roots of unity (since one of the factors of your polynomial is the 3rd cyclotomic polynomials).

$\mathbb{Q}(1)$ is a trivial extension, so you will never really need to write that, and the splitting field by definition should be $\mathbb{Q}(\frac{-1+\sqrt{3}}{2},\frac{-1-\sqrt{3}}{2},e^{\frac{2\pi i}{3}},e^{\frac{4\pi i}{3}})$ by just adding in all the roots, however you can get that by just looking on wolfram alpha or whatever you like.

The content of the question comes in simplifying the above expression since the first two roots are obviously $\mathbb{Q}$ linear combinations of each other. How do the other two roots relate? Can you simplify it any further from there?