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.

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