Field extension of $\mathbb{Q}$ is the splitting field of two different polynomials?

Question

On the topic of splitting fields at Wolfram Mathworld, one of the examples makes sense but the other doesn't. From https://mathworld.wolfram.com/SplittingField.html:

For example, the extension field Q(sqrt(3)i) is the splitting field for x^2+3 since it is the smallest field containing its roots, sqrt(3)i and -sqrt(3)i. Note that it is also the splitting field for x^3+1.

The first one makes sense, but I can't figure out how $\mathbb{Q}(\sqrt 3i)$ is the splitting field of $x^3+1$. Any ideas are appreciated.

Answer 1

Write

\begin{align*} \omega &= \exp\big(\frac{2 \pi i}{6}\big) \\ & = \frac{1}{2} + \frac{\sqrt{3}}{2}i. \end{align*} Then $\omega \in \mathbb{Q}(\sqrt{3}i)$ and $x^3 + 1 = (x+1)(x-\omega)(x+\omega)$ splits in this field.

Answer 2

Hint: Write out the three roots of $x^3+1$