Solution
= Solution
Let $\alpha=2^{1/6}=\sqrt2/\sqrt[3]2$. Then $L=\mathbb Q(\alpha,\zeta_3)$ is the splitting field of $X^6-2$. Eisenstein gives $[\mathbb Q(\alpha):\mathbb Q]=6$, and adjoining nonreal $\zeta_3$ doubles it, so the degree is 12 and the extension is Galois. Valuation at a prime above 2 forces $n\mid6$ if $x^n=2$; conversely $x=\alpha^{6/n}$ works for $n=1,2,3,6$.