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I'm working through an assignment and need to write several elements as polyomials of degree <= 2.

($x^2+1$)($x+1$)

within the field

$\mathbb Z$$_3$[x] / ($x^3 + 2x + 1$)

And am unsure how exactly to go about doing this - could anyone give me some pointers as to what the question is asking me to do?

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    If you have to do this a lot, it really pays to learn to use a "rewrite rule" as Bill Dubuque aptly calls it in his answer. Think about it this way. When you do arithmetic with complex numbers, you always rewrite $i^2=-1$. Here you do the same thing, and rewrite $x^3=-2x-1=x+2$. Consequently also $$x^4=x\cdot x^3=x(x+2)=x^2+2x$$ and so forth (much like $i^3=i\cdot i^2=i(-1)=-i$.) If you want to see several examples of this in the finite field $\mathbb{Z}_2[x]/(x^3+x+1)$, see [the bottom half of my answer to this question.](http://math.stackexchange.com/a/76136/11619)2012-03-29

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