I'm trying to solve this problem:
Show that $p(x)=x^3-2 \in \mathbb Q[x]$, is irreducible over $\mathbb Q$. Find an extension $K$ of $\mathbb Q$ having all roots of $p(x)$ such that $[K:\mathbb Q]=6$.
The easiest part is to show that $p(x)$ is irreducible, it suffices to use the Eisenstein criterion.
By Kronecker's theorem there exists a root $u$ of $p(x)$ in an extension $E$ of $\mathbb Q$, then we can write $p(x)$ as $p(x)=(x-u)q(x)$, where $q(x)\in \mathbb Q[x]$ has degree 2. I know also there is a root $v$ of $q(x)$ in an extension $G$ of $E$.
I noticed if I prove $[\mathbb Q(u,v):\mathbb Q(u)]=2$, where $\mathbb Q(u,v)=\mathbb Q(u)(v)$ I solve the problem, but I don't know how to prove this. This $[\mathbb Q(u,v):\mathbb Q(u)]$ in this context seems very weird to me, I really need help