# Finite orbits

Suppose that $E/F$ is an algebraic field extension, and that $F=E^G$ for some subgroup of automorphisms of $E$ (that is, the extension is Galois). Show that if $e\in E$ then the orbit of $e$ under $\mathrm{Gal}(E/F)$ must be finite. Show that $G$ need not be finite, and show that this fails if the extension is not assumed to be algebraic.

Suppose $G$ is a finite group of automorphisms of a field $E$. Form the ring $E(G)$ which is the subring of the ring of endomorphisms of $E$ generated by $G$ and multiplication by elements of $E$. Note that if $\lambda\in E$ then the composite $g\cdot \mu_\lambda$ equals the composite $\mu_{g(\lambda)} \cdot g$. Show that the $E$-subalgebras of the group ring $E(G)$ are those of the form $E(H)$ for $H$ a subgroup of $G$.
Note By an $E$-subalgebra we mean those subrings that contain all the multiplications by elements $\lambda \in E$.