Let be finite dimensional splitting fields over a field and contained in a field . Show that and are also finite dimensional splitting fields over (and contained in ).

Say and are splitting fields over for the finite sets .

We claim that is a splitting field over for . To see this, note that each polynomial in splits over . Now if is a field over which split, then the polynomials in split over , so that . Likewise , and so . So is a splitting field for over .

Now suppose is irreducible over with a root in . This root is in the splitting field , so splits over by this previous exercise. Likewise, splits over . Thus splits over . Again using this exercise, is a splitting field over .