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$x^5 +2x^2 +1$ 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 $GF(3)$ ?

asdq
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Zoran Mladenovski
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2 Answers2

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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?

  • If GF(3) then I could only input numbers 0,1,2 into my polynomial function ? Only P(0), P(1) , P(2) =0. But in my case none of this equal zero ... P(2) = x^3 -x is not zero, how come ? – Zoran Mladenovski Jan 14 '18 at 01:36
  • Your polynomial is what it is at $0,1,2$, but $0^3+2\cdot0=0$, $1^3+2\cdot 1=0$ and $2^3+2\cdot 2=0$ in GF(3) so you can add $q=x^3+2x$ or any multiple of it to your polynomial without changing its polynomial function. –  Jan 14 '18 at 04:44
  • @ZoranMladenovski So,try out the polynomial above ($x^5+x^3+2x^2+2x+1$) where I've added $q$ to your polynomial, and convince yourself that the polynomial function here is the same as $P$. –  Jan 14 '18 at 07:00
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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 $1$.

asdq
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