Describe the splitting field $F$ of the polynomial $f=x^4+x^2+1$ over $\mathbb{Q}$

Question

I know that $x^4 + x^2 + 1 = (x^2+x+1)(x^2-x+1)$ so the roots of $f$ are $\frac{1\pm \sqrt{-3}}{2}$ and $\frac{-1\pm \sqrt{-3}}{2}$. I am just having trouble writing this in terms of $\mathbb{Q}(x_1,x_2,..)$. Would it just be $\mathbb{Q}(\sqrt{-3})$ since $\frac{1}{2} \in mathbb{Q}$?

Answer 1

Yes, the splitting field is $\mathbb{Q}(\sqrt{-3})$. You already know the roots in $\mathbb{C}$, which are expressible as elements of $\mathbb{Q}(\sqrt{-3})$, so all that really remains to show is that no smaller field extension will split the polynomial. This follows because the roots are not in $\mathbb{Q}$, and $[\mathbb{Q}(\sqrt{-3}):\mathbb{Q}]=2$, so there are no intermediate fields.