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**ThePerfectHacker** Notice that the minimal polynomial of $\displaystyle \sqrt[3]{2}$ over $\displaystyle \mathbb{Q}$ is $\displaystyle x^3 - 2$. The roots of this polynomial is $\displaystyle \sqrt[3]{2},\zeta\sqrt[3]{2},\zeta^2\sqrt[3]{2}$. Thus, the splitting field of this polynomial is $\displaystyle E=\mathbb{Q}(\zeta,\sqrt[3]{2})$. Notice that $\displaystyle E/\mathbb{Q}$ is Galois with $\displaystyle \mathbb{Q}(\sqrt[3]{2})\subset E$ and it is the smallest extension because it is splitting field of the minimal polynomial. Now we see that $\displaystyle \text{Gal}(E/\mathbb{Q}) = S_3$.