MHB List the elements of the field F = F3[x] / < x2 + 1 >

[MupptMath]

Can someone please explain the deconstruction and elements of this set. I understand it to be..

F3[X] = {f(x)=a(0)+a(1)X+...+a(n)X^n : a(i) in F3]
<x^2+1> = {g(x)(x^2+1): g(x) in F3[x]]

So an element in the quotient should be something like f(x)+<x^2+1>

Yet, research shows there are nine elements:
[0], [1], [2], [x], [x+1], [x+2], [2x], [2x+1], [2x+2]

I just don't see how they are derived.

Thanks!

 

[Deveno]

Write $f(x) = q(x)(x^2 + 1) + r(x)$ using polynomial long division.

What can you say about the maximum degree of $r(x)$?

Show $r(x) + \langle x^2 + 1\rangle = f(x) + \langle x^2 + 1\rangle$.

This entails showing $f(x) - r(x)$ is a multiple of $x^2 + 1$.

Does this give you a hint?