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 $\sigma$.

>><b>Question.</b> Is $K$ a finite-dimensional $F$-algebra ?

<b>Remarks. (1)</b>  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.

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

<b>Edit.</b> The following question is motivated by the question: is the evaluation map from $R\{x\}$ to $R^R$ that maps a difference polynomial  $\delta$ to the function $\delta_R:R\rightarrow R$ is injective, at least when $[K:F]$ is infinite?