Let $F$ be a field of characteristic $0$,contained in its algebraic closure $A$. Let $σ$ be an automorphism of $A$ over $F$ and Let $K$ be the fixed field. Prove that every finite extension of $K$ is cyclic.
I could solve like below.
Let $L/K$ be a finite extension , then $[L:K]=2$. So $L/K$ is cyclic.
Is it correct?
Maybe above proof is not correct I can't say$[L:K]=2$.
I started to solve it again. Let $L/K$ be a finite Galois extension $(L \subset A)$.Then $K$ is a fixed field of $<σ>$. So $L/K$ is cyclic. Let $M/K$ be a finite extension. Let $N/K$ be a Galois closure of $M/K$.$M/K$ is cyclic, so $N/K$ is cyclic. Q.E.D.
Is it correct?