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how can I divide for example $\frac{x^2+1}{2x+1}$ in $\frac{\mathbb{Z}_3[x]}{x^3+1}$? It's like normal polynomial dividing but here I got in first step $\frac{x}{2} (\frac{x^2}{2x}=\frac{x}{2}$).What is the modulo value of this statement? I need it for solve a linear equation solve, it's part of the Euclidan algorithm..but here I am stuck and I don't know what to do..Right answer is 2x+2 with remainder 2..But I don't know WHY:D

Please help..And please be patience for my poor English and absence of right formatting, today I've an exam so I am a bit hurry:D

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    $Z_p$ is a field for prime $p$, so every non-zero element will have an inverse. If $p$ is not a prime, you need the element to be coprime to $p$2011-01-03

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HINT $\rm\ \ 3 \equiv 0\ \Rightarrow\ 1 + 2\ x \:\equiv\: 1-x\:.\ $ Now $\rm\ x^3\:\equiv\:-1\ \Rightarrow\ x^2+1\ \equiv\ x^2 -x^3\ \equiv\ x^2\ (1-x)\ $

Therefore $\rm\ (x^2+1)/(1-x)\ \equiv\ \ldots$