Let $K$ be any field, and assume that $K$ satisfy a non-trivial 'generalised' (one variable) polynomial identity involving iterated powers of a given field morphism $\sigma:K\rightarrow K$. Let $F\subset K$ be the subfield fixed by $\sigma$.

Question. Is $K$ a finite-dimensional $F$-algebra ?

Remarks. (1) The answer is yes if the identity is of 'degree 1', i.e. of the form $$\sum_{k=0}^n a_k\sigma^k(x)=a_{{-1}}$$ for some $a_{-1},a_0,\dots,a_n\in K$, and conversely, a finite dimensional $F$-algebra satisfies such an identity.

(2) It is also yes if $char K=p>0$ and $\sigma$ is the Froebenius...