Let $L=\mathbb{C}(X,Y,Z)$ be the rational function field over the complex field and $\sigma$ be automorphism of $L$ over $C$, $$ \sigma(X)=Y,\sigma(Y)=Z,\sigma(Z)=X$$ Moreover let $M$ be the intermediate field of the extension $L/\mathbb{C}$ fixed by the group $<\sigma>$. I think the degree of filed extension $L/M$ is $3$ since the order of group $<\sigma>$ is $3$. But is it true? I know this is true if the extension $L/C$ is finite degree Galois extension. But now the extension $L/C$ is infinite degree extension. So I don't know it is true.
Please give me some advice.
Yes, it is a theorem that if $L$ is a field, and $G$ a finite group of automorphisms of $L$, then $M=L^G$, the elements of $L$ fixed by all elements of $G$, has the property that $L/M$ is a finite Galois extension with Galois group $G$.
See any text on Galois theory for the proof.