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?


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 $1 - 2h(x)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.