I know this has been asked previous on Stackexchange, but I just want to have something clarified:

I am supposed to find an extension K of Q having all roots of $x^3 - 2$ such that $[K:\mathbb Q] = 6.$

Setting $\alpha = 2^\frac{1}{3}$ and $\omega = e^\frac{2\pi i}{3}$, I can write all the roots in a basis such that the degree is $3$: $$\{\alpha,\alpha \omega, \alpha \omega^2\}$$ I guess this is a field of degree 3? So I am then supposed to add three more random linear independent elements such that $[K:\mathbb Q] = 6$?