Question

Show that 1+3x is a unit in ${\mathrm{\mathbb{Z}}}_9\left[x\right]$. Hence, Corollary 4.5 may be false if R is not an integral domain.

Short answer

It is proved that 1+3x is a unit in ${\mathrm{\mathbb{Z}}}_9\left[x\right]$.

Step-by-step solution

Polynomial Arithmetic

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

Fields:

The given unit is 1+3x . The multiplicative inverse of this will be: 1-3x.

Then, we have:

$\left(1+3x\right)\left(1-3x\right)=1-9x^2=1$

Clearly found to be in ${\mathrm{\mathbb{Z}}}_9\left[x\right]$.

Hence proved, 1+3x is a unit in ${\mathrm{\mathbb{Z}}}_9\left[x\right]$.