abstract algebra - Two number fields with trivial intersection, not linearly disjoint but....

Do there exist two finite extensions of $\mathbb{Q}$, $L=\mathbb{Q}(\alpha)$ and $K=\mathbb{Q}(\beta)$ such that $L\cap K=\mathbb{Q}$, the minimal polynomial $\mu_{\alpha}(X)\in \mathbb{Q}[X]$ of $\alpha$ over $\mathbb{Q}$ is not irreducible over $K$, but $\mu_\alpha(\beta)\neq 0$ ?

Answer

Sure. Let $\alpha=\root3\of2$, let $\gamma$ be a nonreal cube root of 2, and let $\beta=\gamma+1$.