Let be an algebraic field extension and let be a subring containing . Show that is a subfield of .
It suffices to show that is closed under inversion. To this end, let . Since is algebraic over , every element of is algebraic over . Let be the minimal polynomial of over ; if , then we have . Note that since is irreducible. Rearranging, we see that , and . So , and is closed under inversion. Thus is a subfield of .