Let $K_1$ and $K_2$ be finite extensions of $F$ contained in the field $K$, and assume both are splitting fields over $F$.

\begin{enumerate}[(a)]

\item Prove that their composite $K_1 K_2$ is a splitting field over $F$.

\item Prove that $K_1 \cap K_2$ is a splitting field over $F$. Use the preceding exercise.

\end{enumerate}

\subsubsection*{Solution}

\begin{enumerate}[(a)]

\item Let $K_1$ be the splitting field of $f_1 \in F[x]$ and $K_2$ be the splitting field of $f_2 \in F[x]$. Then $K_1 K_2$ contains the roots of both $f_1$ and $f_2$. Hence $K_1 K_2$ is the splitting field over $F$ of the polynomial $g = f_1 f_2$.

\item Let $f \in F[x]$ be an irreducible polynomial with a root in $K_1 \cap K_2$. Then both $K_1$ and $K_2$ contain a root of $f$. By the previous exercise, $f$ splits completely in these fields – they contain at least one, hence all of the roots of $f$. Therefore $f$ must also split completely in their intersection.