A not constant unit in the ring $\mathbb{Z}/27\mathbb{Z}[X]$ and another one in $\mathbb{Z}/96\mathbb{Z}[X]$.

Question

Could someone help me to find a unit (not constant) in the ring $\mathbb{Z}/27\mathbb{Z}[X]$ ? And another one in the ring $\mathbb{Z}/96\mathbb{Z}[X]$ ?

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I have done the following:

Let we take a polynomial $f\in\mathbb{Z}/27\mathbb{Z}[X]$. $f$ will be a unit if there exists another polynomial $g\in\mathbb{Z}/27\mathbb{Z}[X]$ such that $f\cdot{g} =1$.

I also know that the polynomials from $\mathbb{Z}/27\mathbb{Z}[X]$ have their coefficients in $\mathbb{Z}/27\mathbb{Z}=\{\bar0,\bar1,\bar2,...,\bar{26}\}$ and $\bar{a}=\bar{b}\leftrightarrow a\equiv b\pmod {27} $

But how can I follow now?

Answer 1

Hint. Let $R$ be a ring and $a \in R$ be nonzero nilpotent, i.e., let $a \neq 0$ be such that $a^n = 0$ for $n \geqslant 1$.
Consider the polynomial $f(X) = 1 - aX \in R[X]$. What happens if you multiply with $1 + aX$? What if you multiply that with $1 + a^2X^2$? Then with $1 + a^4X^4$?...

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A more general result:

A polynomial in $R[x]$ is a unit iff the constant coefficient is a unit in $R$ and the other coefficients are nilpotent.