3
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Related. $\endgroup$ – José Carlos Santos Jul 10 at 21:59
  • 2
    $\begingroup$ 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. $\endgroup$ – David C. Ullrich Jul 10 at 22:05
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.