Special polynomials

**I-Think** · November 14th 2010, 06:48 AM

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.
Can anyone help out please?

**roninpro** · November 14th 2010, 07:24 AM

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

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

**I-Think** · November 14th 2010, 11:29 AM

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 $\mathbb{Z}_2$

**Drexel28** · November 14th 2010, 11:37 AM

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?

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