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 $\mathbb{Z}/96\mathbb{Z}[X]$ ?
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?