Show that E is a finite extension of F

Question

Given that E is a field extension of a field F, and E is a splitting field of F of some nonconstant polynomial $f(x) \in F[x]$ .

How do I show that E is a finite extension of F?

Answer 1

$f$ has finitely many zeros $\alpha_1,\ldots,\alpha_n$ in $E$. Then $E=F(\alpha_1,\ldots,\alpha_n)$. Then each extension $F(\alpha_1,\ldots,\alpha_k)/F(\alpha_1,\ldots,\alpha_{k-1})$ is simple and generated by an algebraic element, so is finite. As $F(\alpha_1,\ldots,\alpha_n)/F$ is a finite tower of finite extensions, it is also finite.