# Formal derivative vs Little Fermat

Are formal derivatives studied in finite fields? If so, what is wrong or what would be the justification for the following: Let $$\partial$$ denote the formal derivative of a polynomial over $$\mathbb{F}_p$$ for $$p$$ prime. Then, thanks to little Fermat, we have that $$X^p \equiv_p X$$, allowing us to conclude $$\partial X^p \equiv_p \partial X \equiv_p 1$$. At the same time, we have $$\partial X^p \equiv_p p X^{p-1} \equiv_p 0$$.

In a polynomial ring $$R[X]$$, $$X$$ is not a number. So $$X^n = X$$ does not hold. In general, equations from the base ring $$R$$ will not hold on $$X$$. Just the basic axioms of rings like $$X^a X^b = X^{a+b}$$.

• $$\mathbb F$$ is a field and
• $$(x_i)_{i \in \mathbb N}$$ is a sequence of pairwise different elementss of $$\mathbb F$$ and
• $$p$$ is a polynomial over $$\mathbb F$$ and
• $$p(x_i)=0, \forall i \in \mathbb N$$
then $$p = 0$$ (the null polynomial)
From this follows that if $$\mathbb F$$ is a field that is not finite, like $$\mathbb{ Q,R,C}$$ and $$p(x)=0, \forall x \in \mathbb F$$ then $$p=0$$. But this cannot be concluded for finite fields $$\mathbb F$$.