1. ## Special polynomials

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

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

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

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

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

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

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

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

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