I am having hard time to solve following exercise.
Let $\Omega$ be the algebraic closure of a field $k$.
a) Suppose that every finite extension of $k$ is cyclic. Prove that it exists $\sigma \in Gal(\Omega / k)$ such that $k=\{x \in \Omega : \sigma(x)=x\}$.
b) For $\tau \in Gal(\Omega / k)$ and $K=\{x \in \Omega : \tau(x)=x\}$, prove that every finite extension of $K$ is cyclic.
For a) I'm trying to use Zorn lemma, but I'm going really nowhere.