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Let's take a polynomial $m(x)$ from $\mathbb{Z}_{3}[x]$. Now, $\mathbb{Z}_{3}$ should contains the integers $-1,-2,-3$. However after reading few exercises about this argument i suspect that we can ignore negative values when we work modulo $m(x)$. But, why ? (Intuitively)

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  • $\begingroup$ What is your definition of $\,\Bbb Z_3[x]?\ \ $ $\endgroup$ May 23, 2019 at 15:50

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$\mathbb Z_3=\{[0],[1],[2]\}$ where $$[n]=\{n+3k|k\in\mathbb Z\}$$ which means that $[-3]=[0], [-2]=[1]$ and $[-1]=[2]$.

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  • $\begingroup$ What i don't understand is why the polynomials with negative coefficients are equal to ones with positive coefficients ... $\endgroup$
    – AleWolf
    May 23, 2019 at 8:37
  • $\begingroup$ @AleQuercia Because there is no such thing as "negative" and "positive" in $\mathbb Z_3$. The number $-2$ is not an element of $\mathbb Z_3$. The equivalence class of this number is an element. And this class contains both positive and negative numbers. $\endgroup$
    – 5xum
    May 23, 2019 at 9:31
  • $\begingroup$ @5xum Keep in mind that many ENT textbooks use only congruences, not quotient rings. $\endgroup$ May 23, 2019 at 15:48
  • $\begingroup$ I think that what you wrote derives from the fact that when working with polynomials modulo negative coefficients are equivalent to positive ones $\endgroup$
    – AleWolf
    May 23, 2019 at 16:38

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