Units of Ring of Polynomial Forms over Commutative Ring

Theorem
Let $\left({A, +, \circ}\right)$ be a commutative ring with unity whose zero is $0_A$ and whose unity is $1_A$.

Let $A \left[{X}\right]$ be the ring of polynomial forms in an indeterminate $X$ over $A$.

Let $P\left({X}\right) = a_0 + a_1 X + \cdots + a_nX^n \in A \left[{X}\right]$.

Then $P\left({X}\right)$ is a unit of $A \left[{X}\right]$ if and only if $a_0$ is a unit of $A$, and for $i = 1,\ldots,n$, $a_i$ is nilpotent in $A$.

Corollary
Let $A$ be a reduced ring.

Let $A \left[{X}\right]$ be the ring of polynomial forms in an indeterminate $X$ over $A$.

The group of units of $A \left[{X}\right]$ is precisely the group of elements of $A \left[{X}\right]$ of degree zero that are units of $A$.