Exercise 11, § 11

Let K be a field whose algebraic closure $\Omega$ is an extension of finite degree $> 1$ of K. Show that $\Omega = K(i)$ where $i^2 = -1$. (If we had $\Omega \neq K(i)$, show with the help of Galois theory and Sylow's theorem that there would then exist a field E such that $K(i) \subset E \subset \Omega$ and $\Omega$ is an extension of prime degree of E; now apply Ex. 10.)