Let $(K,\sigma)$ be a *difference field* of characteristic $0$, *i.e.* equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $\delta=0$ for some $\delta\in K\{x\}$, where $$K\{x\}=K\left[\sigma^i(x):i\in\mathbb N\right].$$ Let $F=\{x\in K:\sigma x=x\}$ be the subfield fixed by $\sigma$.



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> > **Question.** Is $[K:F]$ finite?
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**Remark.**  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.


**Edit.** The following question is motivated by the question: 



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> > **Question.** Is the evaluation map that maps a difference polynomial $\delta\in K\{x\}$ to the function $\delta\_K:K\rightarrow K$ is injective, at least when $[K:F]$ is infinite?
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On difference identities and $[K:F]$
The answer is yes and appears in R. Cohn's book 'Difference algebra' Lemma II p. 201.


