Suppose that is an algebraic field extension, and that for some subgroup of automorphisms of (that is, the extension is Galois). Show that if then the orbit of under must be finite. Show that need not be finite, and show that this fails if the extension is not assumed to be algebraic.
Suppose is a finite group of automorphisms of a field . Form the ring which is the subring of the ring of endomorphisms of generated by and multiplication by elements of . Note that if then the composite equals the composite . Show that the -subalgebras of the group ring are those of the form for a subgroup of .
Note By an -subalgebra we mean those subrings that contain all the multiplications by elements .