In Linear Algrebra form Hoffman and Kunze, the Taylor's Formula is stated as follows:
Theorem 5. (Taylor's Formula) (page 129) Let $\mathbb{F}$ be a field of characteristic zero, $c\in \mathbb{F}$, and $n$ a positive integer. If $f$ is a polynomial over $\mathbb{F}[X]$ with $\deg f \leq n$, then $f=\sum_{k=0}^{n}\frac{(D^{k}f)}{k!}(c)(x-c)^{k}.$
After proving the Theorem, they make some comments."Although we shall not give any details, it is possible worth mentioning at this point that with the proper interpretation Taylor's Formula is also valid for polynomials over fields of finite characteristics. If the field $\mathbb{F}$ has finite characteristics then we may $k!=0$ in $\mathbb{F}$, in which case the division of $(D^{k})f(c)$ by $k!$ is meaningless. Nevertheless, sense can made out of the division of $D^{k}f$ by $k!$, because every coefficient of $D^{k}f$ is an element of $\mathbb{F}$ multiplied by an integer divisible by $k!$."
Is this the "proper interpretation"? I was hoping an interpretation over a field like $GF(2)$.
Thanks for your help.