How would I work out the Cayley table for $F_3 [x]$ modulo $x^2 +2$ with addition and multiplication.

Question

How would one display the Cayley table for $F_3 [x]/(x^2 +2)$ and show that it is a ring (I have assumed addition and multiplication are associative and that multiplication is distributive over addition).

Answer 1

Hint:

In $\mathbf F_3[x]$ , $x^2+2=x^2-1$, hence by the Chinese remainder theorem, the quotient is isomorphic to $$\mathbf F_3[x]/(x-1)\times\mathbf F_3[x]/(x+1)\simeq\mathbf F_3\times\mathbf F_3.$$

Answer 2

If you are looking for a less powerful approach than Bernard's answer, you can just write down each element of $\frac{\mathbb{F}_3[x]}{x^2+2}$. $$\frac{\mathbb{F}_3[x]}{x^2+2} = \{ a_0 + a_1 \alpha \ | \ \alpha^2=-2 = 1 \textrm{ and } a_0,a_1\in \mathbb{F}_3\}.$$ When you take a quotient of a ring by an ideal, you always get a ring. But you can write down the multiplication table if you wanted to.

For example: $(1 + \alpha)*\alpha = \alpha + \alpha^2 = \alpha + 1$.

Another example: $(1 + 2\alpha)*(2+\alpha) = (2 + \alpha + 4\alpha + 2\alpha^2) = 2 +5\alpha + 2\alpha^2 = 2 + 2\alpha + 2 = 4 + 2\alpha = 1 + 2\alpha.$