 Finite extensions of the fixed field of a finite-order automorphism of an algebraically closed field

Let an automorphism $\sigma$ of an algebraically closed field $L$ have finite order and fixed field $K$. Artin's fixed-field theorem makes $L/K$ finite cyclic Galois. Every finite extension $M/K$ embeds in $L$: in characteristic zero, choose a primitive generator and send it to a root in $L$. Its image is an intermediate field of a cyclic extension and is therefore Galois and cyclic over $K$.