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The units of $\mathbb Z_4[x]$ should be $$\{f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n\mid a_0\in\{1,3\},a_i\in\{0,2\}, 1\le i\le n\}.$$ But I don't know how to prove. Any suggestion?

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Maybe you even want to proof the general result:

For any commutative Ring $R$ a polynomial $\sum_{k=0}^n a_k X^k$ is a unit in $R[X]$ if and only if $a_0$ is a unit in $R$ and $a_k$ is nilpotent for each $1 \leq k \leq n$.

Note that one direction is immediately clear, since any sum of a unit and a nilpotent is again a unit.

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