What is the splitting field of $x^3 - \pi$? Is it $\mathbb R(\sqrt[3] \pi, \xi_3)$ or $\mathbb Q(\sqrt[3] \pi, \xi_3)$? (where $\xi_3$ denotes the third root of unity)
It is a polynomial over $\mathbb R[x]$, so I guess it must be $\mathbb R(\sqrt[3] \pi, \xi_3)$, but I never saw such an extension.
Since $\sqrt[3]\pi$ is already an element of $\mathbb R$ and $\xi_3=-\frac12\pm i\frac{\sqrt{3}}2$, the splitting field is simply $\mathbb C$. In fact, $\mathbb R$ and $\mathbb C$ are the only candidates for algebraic extensions of $\mathbb R$.