Question: Suppose that $K/k$ is a field extension, and $K$ is algebraically closed. Let $k \subseteq A \subseteq K$ and $k \subseteq B \subseteq K$ with $A/k$ and $B/k$ normal extensions. Show that $A \cap B/k$ is a normal extension.
Attempt: Let $f\in k[x]$ be an irreducible polynomial that has a root, say $\alpha$, in $A \cap B$. To show normality, we must show that $f$ splits over $A \cap B$. Clearly $\alpha \in A$ so by normality $f$ must split over $A$. Similarly with $B$. Then all of $f$'s roots are in both $A$ and $B$ and so they are all in $A \cap B$ as well. Hence $f$ splits over $A \cap B$ as required.
My attempt seems too simplistic to be right, and I haven't used $K$ at all, which makes me think I'm missing something. If I've gone wrong somewhere I'd really appreciate a push in the right direction.