<s>[INST] Trouble constructing $\mathbb Z_3[x]/(x^2+1)$ If I have $\mathbb Z_3[x]/(x^2+2x+2)$, I can construct a field by letting $x^2=x+1$. The reps are:
$0$
$1$
$x$
$x^2=x+1$
$x^3=x^2+x=x+1+x=2x+1$
$x^4=2x^2+x=2x+2+x=2$
$x^5=2x$
$x^6=2x^2=2x+2$
$x^7=2x^2+2x=2x+2+2x=x+2$

However, when I try this for $\mathbb Z_3[x]/(x^2+1)$, letting $x^2=-1=2$, it does not work. $x^2+1$ is irreducible over $\mathbb Z_3$ and thus we know $\mathbb Z_3[x]/(x^2+1)$ is a field. 
Here's what I get
$0$
$1$
$x$
$x^2=2$
$x^3=2x$
$x^4=2x^2=4=1$
Why doesn't this work with $x^2+1$?
 [/INST] If I understand you correctly, the issue is that there's no guarantee that $x$ will be a multiplicative generator for the group of units for the resulting field.  You might to choose another element.  One choice might be $x+2$, which squares to $x^2+4x+4 \equiv x$.
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