Quotient group members $\mathbb{Z}_3[X]/N$ with $N=1+X^2$

Question

What are the members of the quotient group $\mathbb{Z}_3[X]/N$ with $N=1+X^2$?

I do not know how to write this out. Which operation do you use? Are these all possible polynomials in $\mathbb{Z}_3[X]$ multiplied or added with $N$?

Answer 1

You take a polynomial $f\in\mathbb{Z}_3[X]$ and divide it with remainder by $X^2+1$. The remainder is your result.

For example: $2X^3+2X+1\in\mathbb{Z}_3[X]$, then $(2X^3+2X+1)\div (X^2+1)= 2X~~ R~~ 1$ (R stands here for remainder)

It is $2X^3+2X+1=2X(X^2+1)+1$, hence $[2X^3+2X+1]=[1]$.