Polyomial function over ring GF(3)

Question

$x^5 +2x^2 +1$ is a polynomial over ring $GF(3)$ and let $P(x)$ be its polynomial function ... Is there any other polynomial over the same ring that corresponds to the same polynomial function?

I've read in the book that it exists, but I do not understand. Would somebody be willing to explain to me and give me an example of another polinomial over the same ring $GF(3)$ ?

Answer 1

In $GF(3)$ you have $x^3-x=0$ for all $x$. Thus, the polynomial function for the polynomial $q=x^3-x=x^3+2x$ is $0$. Also, any multiple $qr$ of $q$ also has all-zeros polynomial function, since $(qr)(x)=q(x)r(x)=0\cdot r(x)=0$.

Now back to your example: add to your polynomial any multiple of $q$: this will give you plenty of polynomials that have the same polynomial functions as $P(x)$. E.g. $(x^5+2x^2+1)+(x^3+2x)=x^5+x^3+2x^2+2x+1$ is one such polynomial.

Exercise: divide your polynomial by $q$ and the remainder will again have the same polynomial function. What is it?

Answer 2

If you evaluate $P$ at each $x\in GF(3)$ you will find $P(x)=1$ for every such $x$. Therefore as a function on $GF(3)$ $P$ is the same as the constant $1$.