# When notion of polynomial and polynomial function coincide.

Before I never thought about the difference between a polynomial function and polynomial over a certain field $$K$$. Given examples such as when $$K = R$$ or $$K = C$$, we see that sometimes the notion of polynomial and polynomial function coincide in a sense that if $$f(x) = a_nx^n+...+a_0$$ is a polynomial function, then $$f\equiv 0 \iff a_n = a_{n-1} = ... = a_0 = 0.\tag{1}$$ Sometimes it does not coincide, as we can see when $$K = Z_2$$.

My question is, what are fields for which the notion of polynomial and polynomial function like in $$(1)$$ coincide called, and what are their basic properties?

I suspect that $$(1)$$ never holds for a finite field.

• – José Carlos Santos Jul 10 at 21:59
• You already have an answer, showing that this happens if and only if the field is infinite. You might note that it's obviously false for a finite field, since then there are only finitely many functions, hence finitely many polynomial functions. – David C. Ullrich Jul 10 at 22:05

## 1 Answer

They coincide iff the field is infinite. Over any finite field whose elements are $$r_1,\dots,r_n$$, note that the polynomial $$(x-r_1)(x-r_2)\dots(x-r_n)$$ vanishes identically but its coefficients are not all $$0$$ since the leading coefficient is $$1$$. On the other hand, over any infinite field, if a polynomial of degree $$n$$ cannot have more than $$n$$ roots (since each root gives a linear factor) and so a polynomial with infinitely many roots can only be the zero polynomial.