How can I find all elements of this field?

Question

I am given this finite field.

$$K=\mathbb{Z_3}[x]/p$$ where $p(x)=x^2 + 1$

How do I find all elements of this field?

Answer 1

Well, first we note (using that $x^2 + 1$ is degree 2) since it has no roots (which we can check by plugging in $-1, 0$, and $1$) it is irreducible, so since $\mathbb{Z}_3[x]$ is a PID we have $\mathbb{Z}_3[x]/(p)$ is a field.

Now we see that, since we are "moding out" by a two degree polynomial (ideal) we have that elements of $\mathbb{Z}_3[x]/(p)$ can only be $0$ and $1$ degree polynomials; i.e. elements of the form $a + bx$; $a,b \in \mathbb{Z}_3$.

Answer 2

Elements of the field are the following cosets $0,1,2,x,2x,x+1,x+2,2x+1,2x+2.$