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Compute the units in $\mathbb{Z}_4[x]$.

My Work:

I have seen a problem to show the following statement (So, it is not a theorem):

$R$-commutative ring with identity. Then $a_0+a_1x+\ldots +a_nx^n=p(x)\in R[x]$ is a unit iff $a_0$ is a unit and $a_1,...,a_n$ are nilpotent.

I can prove this but it is a very long proof. Also when I prove it I can use it to solve the above problem. My question is, is there a simple way of computing the units of $\mathbb{Z}_4[x]$, without using above claim?

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If $fg = 1$, then modulo 2 we have $\bar f\bar g = 1$ in $\Bbb F_2[x]$, so $\bar f = 1$. It follows that $f = 1 + 2h(x)$ and this always is invertible with inverse $1 - 2h(x)$.

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