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.

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Finite orbits

Galois correspondence via group algebras

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.

Galois correspondence via group algebras