## Composites and intersections of finite dimensional splitting fields are splitting fields

Let $K_1,K_2$ be finite dimensional splitting fields over a field $F$ and contained in a field $K$. Show that $K_1K_2$ and $K_1 \cap K_2$ are also finite dimensional splitting fields over $F$ (and contained in $K$).

Say $K_1$ and $K_2$ are splitting fields over $F$ for the finite sets $S_1,S_2 \subseteq F[x]$.

We claim that $K_1K_2$ is a splitting field over $F$ for $S_1 \cup S_2$. To see this, note that each polynomial in $S_1 \cup S_2$ splits over $K_1K_2$. Now if $E$ is a field over which $S_1 \cup S_2$ split, then the polynomials in $S_1$ split over $E$, so that $K_1 \subseteq E$. Likewise $K_2 \subseteq E$, and so $K_1K_2 \subseteq E$. So $K_1K_2$ is a splitting field for $S_1 \cup S_2$ over $F$.

Now suppose $q(x)$ is irreducible over $F$ with a root in $K_1 \cap K_2$. This root is in the splitting field $K_1$, so $q$ splits over $K_1$ by this previous exercise. Likewise, $q$ splits over $K_2$. Thus $q$ splits over $K_1 \cap K_2$. Again using this exercise, $K_1 \cap K_2$ is a splitting field over $F$.