Let $p=x^2+1 \in \mathbb{Z}/3[x]$ . write down the multiplication table for $\mathbb{Z}/3[x]/(p)$

Question

Let $p=x^2+1 \in \mathbb{Z}/3[x]$ . write down the multiplication table for $\mathbb{Z}/3[x]/(p)$

sorry i don't know how to solve that's the reason i am not add any effort plesae any one help me

Answer 1

So we consider $x^2 + 1$ to be equivalent to $0$ in the ring $\mathbb{Z}_3[x]$. So $x^2 \equiv -1 \equiv 2$. So all polynomials can be reduced to at most linear ones, by reducing $x^2$ and also higher powers using these equivalences. We have 3 options for constant terms, and $3$ for the coefficient of $x$, and for all those we have

$$(a + bx)(c+dx) = ac + (bc+ad)x + bdx^2 \equiv (ac-bd) + (bc+ad)x$$

Now you can make a table...(coefficients reduced mod $3$ of course)