11.79 (1990)

Let $G$ be a finite group of automorphisms of an infinite field $F$ of characteristic $p$. Taking integral powers of the elements of $F$ and the action of $G$ define the action of the group ring $\mathbb{Z}G$ of $G$ on the multiplicative group of $F$. Is it true that any subfield of $F$ that contains the images of all elements of $F$ under the action of some fixed element of $\mathbb{Z}G \setminus p\mathbb{Z}G$ contains infinitely many $G$-invariant elements of $F$?