Composite and intersection of splitting fields (D&F 13.4.6)

Let \(K_1\) and \(K_2\) be finite extensions of \(F\) contained in the field \(K\), and assume both are splitting fields over \(F\).
(a) Prove that their composite \(K_1K_2\) is a splitting field over \(F\).
(b) Prove that \(K_1 \cap K_2\) is a splitting field over \(F\).

(a) \(K_1\) and \(K_2\) are splitting fields over sets of polynomials \(I\) and \(J\) respectively, whose roots make up sets \(A\) and \(B\). Note that \(K_1 = F(A)\) and \(K_2 = F(B)\). Let \(P\) be the set of all possible products of the polynomials; then \(K_1K_2\) contains \(A\) and \(B\). Furthermore, there cannot be a proper subfield \(F \subseteq K \subset K_1K_2\) with this property. Otherwise, \(K\) would also contain \(A\) and \(B\), and hence \(F(A)\) and \(F(B)\), contradicting the fact that \(K_1K_2\) is the smallest field containing \(F(A)\) and \(F(B)\).

(b) Let \(f(x)\) be an irreducible polynomial in \(F[x]\) with a root \(\alpha \in K_1 \cap K_2\). If \(f\) is linear, we are done. Otherwise, since \(f(x)\) splits completely in \(K_1[x]\) we can write \(f(x) = (x-\alpha)(x-\beta)g(x)\) for \(\beta \in K_1\). Since \(K_2(\alpha) = K_2 \cong K_2(\beta)\), it follows that \(\beta \in K_2\) as well. By induction we can prove similarly for irreducible factors of \(g(x)\), so that \(f(x)\) factors completely in \((K_1 \cap K_2)[x]\).