# No element of degree $0$ (constant $= a_0 X^0$) in $\mathbb{Z}_4/(2X) \mathbb{Z}_4$

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?

$$X = a X^0$$ 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 $$\,X = 0$$
• I always forget that you can just plug in some $X$, since it must hold for ALL $X$, thanks for reminding me! – Wesley Strik Oct 19 '18 at 7:19