find a field $F$ and different polynomials $f(X),g(X)\in F[X]$ that for every $\alpha \in F$ we have $f(\alpha)=g(\alpha)$.
prove that it is impossible if $F$ is infinite.
i think this example works: $F=Z_2 , f(X)=0 , g(X)=x^2-x $
am i true?
how i can prove it when $F$ is infinite?
thanks
Yes, your example is right and, in general, in $\mathbb{F}_p$, $f(X)=0$ and $g(X)=X^p-X$ are an example of this sort.
If $F$ is infinite, let $d=\max\{\deg f, \deg g\}$ and pick $a_1,\ldots, a_{d+1}\in F$ all distinct (you can, because $F$ is infinite). Then $f(X)-g(X)=p(X)$ is a polynomial with $\deg p\leq d$ and $p(a_i)=0$ for every $a_i$, because you have that $f(a)=g(a)$ for every $a\in F$. But a polynomial, with coefficients in a field, of degree $d$ vanishing for $d+1$ values in such field has to be zero. Then $f\equiv g$.