Let $K$ be any field, and assume that $K$ satisfy a non-trivial 'generalised' 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 $\sum a_n\sigma^n(x)=0$ for some $a_n\in K$, 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...