1. Special polynomials

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.

2. I think that $\displaystyle \mathbb{Z}_p$ is a good place to start.

Why not take $\displaystyle f(x)=x^2+x$ over $\displaystyle \mathbb{Z}_2$?

3. Tried it, but in $\displaystyle \mathbb{Z}_2$, by Fermat's Little Theorem, I believe $\displaystyle x^2\equiv{x} (mod$ $\displaystyle 2)$
So that becomes
$\displaystyle f(x)= x+x=2x$ and $\displaystyle 2\equiv0 (mod$ $\displaystyle 2)$

So that is in fact the zero polynomial in $\displaystyle \mathbb{Z}_2$

4. Originally Posted by I-Think
Tried it, but in $\displaystyle \mathbb{Z}_2$, by Fermat's Little Theorem, I believe $\displaystyle x^2\equiv{x} (mod$ $\displaystyle 2)$
So that becomes
$\displaystyle f(x)= x+x=2x$ and $\displaystyle 2\equiv0 (mod$ $\displaystyle 2)$

So that is in fact the zero polynomial in $\displaystyle \mathbb{Z}_2$
But, by your criterion any polynomial such that $\displaystyle p(c)=0$ for all $\displaystyle c$ is the zero polyommial. In fact, the zero polynomial is the polynomial $\displaystyle p(x)=0+0x+0x^2+\cdots+0x^n$. Now, $\displaystyle p(x)=x^2+x$ does not have all coefficients of $\displaystyle 0$, does it?