Exercise 9, § 15

Let $E$ be a field, $G$ a group of automorphisms of $E$ and $K$ the subfield of $E$ consisting of the elements invariant under $G$. a) Show that for every subfield $L$ of $E$ which is stable for the elements of $G$, $L$ and $K$ are linearly disjoint over $L \cap K$. (Consider a linear relation $\sum \lambda_j x_j = 0$ between the elements x_j \in K which are linearly independent over L \cap K, with $\lambda_j \in L$ not all zero, the number of $A_j \neq 0$ being the least possible.) b) If $L \cap K$ is the field of invariants of a group of automorphisms of K, show that L is the field of invariants of a group of automorphisms of $L(K)$.