Exegesis. why $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })$ is the splitting field [1505BA]

Suppose $K$ is a splitting field of $x^{3} - 2$ in $\mathbb{Q}[x]$. The roots of $x^{3} - 2$ are $$\theta_{1} = \sqrt[3]{ 2 }, \quad \theta_{2} = \sqrt[3]{ 2 } e^{ 2 \pi i / 3 } = \sqrt[3]{ 2 } \frac{-1 + \sqrt{ -3 }}{2}, \quad \theta_{3} = \sqrt[3]{ 2 } e^{ 4 \pi i / 3 } = -\sqrt[3]{ 2 } \frac{-1 + \sqrt{ -3 }}{2}.$$ They all belong to $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })$, hence $K \subseteq \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })$. On the other hand, if $K$ contains all the roots, then $\theta_{1} \in K$ and $\theta_{1}^{2}\theta_{2} = -1 + \sqrt{ -3 } \in K \implies \sqrt{ -3 } \in K$, and $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) \subseteq K$. Therefore, $K = \mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })$.