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Consider $\mathbb{Z}_4[X]/(2X) \mathbb{Z}_4[X]$. Then we wish to show/verify that the residue class $X$ does not contain an element of degree $0$.

In the previous exercise the book asked that if we have a monic polynomial in $R[X]$ then division by this polynomial causes the residue classes to be of lesser degree. Now of course $2X^1$ is not monic, so that does not mean it has to have a constant residue class($X^0$), but in fact we need to show that there are NONE. Why is this the case?

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$X = a X^0$ in $R\!\iff\! X = a X^0\! + (2X) f(X)\,$ in $\,\Bbb Z_4[X]$ $\,\Rightarrow\, a = 0\, $ in $\,\Bbb Z_4,\,$ by evaluation at $\,X = 0$

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  • $\begingroup$ I always forget that you can just plug in some $X$, since it must hold for ALL $X$, thanks for reminding me! $\endgroup$ – Wesley Strik Oct 19 '18 at 7:19

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