Compute the splitting field of x⁴+x²+1 over QQ

Compute the splitting field of $p(x) = x^4 + x^2 + 1$ over $\mathbb{Q}$, and its degree.

Note that $p = g \circ h$, where $g(y) = y^2+y+1$ and $h(x) = x^2$. Evidently, the roots of $g(y)$ are $\eta = \dfrac{-1 \pm i\sqrt{3}}{2}$, and $\zeta$ is a root of $p$ if $h(\zeta) = \eta$. This yields the (distinct) roots $\zeta = \pm \sqrt{\dfrac{-1 \pm i\sqrt{3}}{2}}$. There are four of these, and so we have completely factored $p$.

The splitting field of $p(x)$ over $\mathbb{Q}$ is thus $\mathbb{Q}(\pm \sqrt{\dfrac{-1 \pm i\sqrt{3}}{2}}) = \mathbb{Q}(\sqrt{-1/2 \pm \sqrt{-3/4}})$.

Using this previous exercise, we have $\sqrt{-1/2 + \sqrt{-3/4}} = 1/2 + i\sqrt{3}/2$ and $\sqrt{-1/2 - \sqrt{-3/4}} = 1/2 - i\sqrt{3}/2$.

Evidently, then, the splitting field of $p(x)$ is $\mathbb{Q}(i\sqrt{3})$. Since $i\sqrt{3}$ is a root of the irreducible $x^2+3$ (Eisenstein), this extension of $\mathbb{Q}$ has degree 2.