Exercise 20, § 10

Let E be an algebraic extension of a field K such that every non-constant polynomial in K[X] has at least one root in E. Show that E is algebraically closed. (In an algebraic closure of E, hence of K, let F be an extension of finite degree of K. To show that F ⊂ E reduce to the case where F is quasi-Galois and then, by means of Prop. 13 of V, p. 76, to the case where F is either Galois or p-radical.)