Give an example of a field $\displaystyle F $ and a polynomial $\displaystyle f(x)\in{F[x]}$ which is not the the zero polynomial but $\displaystyle f(c)=0$ for all $\displaystyle c\inF$

I've tried the fields $\displaystyle Z$ $\displaystyle (mod p)$ where $\displaystyle p$ is prime, but now I don't believe these can work , there are $\displaystyle n$ elements in that field, but the highest attainable power is n-2, so that's a dead end, in my understanding.

Fields $\displaystyle Z$, $\displaystyle Q$, and $\displaystyle R$ also don't work, according to my (possibly incorrect) understanding.

Can anyone help out please?