abstract algebra - Show that $4x^2+6x+3$ is a unit in $mathbb{Z}_8[x]$.

Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$.

Once you have found the inverse like here, the verification is trivial. But how do you come up with such an inverse. Do I just try with general polynomials of all degrees and see what restrictions RHS = $1$ imposes on the coefficients until I get lucky? Also is there a general method to show an element in a ring is a unit?

Answer

If $R$ is a commutative ring: the units in $R[x]$ are the polynomials whose constant term is a unit, and whose higher order coefficients are nilpotent. You can apply this directly to your example.