Exercise 9, § 14

An extension E of a field K is called a Dedekind extension of K if for every subextension F of E, F is identical with the subfield of elements of E invariant under the group $\mathrm{Aut}_F(E)$ of all F-automorphisms of E. a) For an algebraic extension E of K to be Dedekind over K it is necessary and sufficient that E should be a Galois extension of K. b) Suppose that K is of characteristic $p > 0$. Show that a Dedekind extension of K is necessarily algebraic over K (hence Galois). (Use Ex. 8, b.) c) If E is a Dedekind extension of K and L a pure transcendental subextension of E, distinct from K and such that E is an algebraic extension of L, show that E has infinite degree over L.