Ring of Polynomial Forms is Integral Domain

Theorem
Let $D$ be an integral domain.

Let $\left\{{X}\right\}$ be a singleton.

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

Then the $D \left[{X}\right]$ is an integral domain.

Proof
By Ring of Polynomial Forms is Ring we know that $D \left[{X}\right]$ is a commutative ring with unity.

Finally we note that if neither $\displaystyle f \left({x}\right) = \sum_{k \mathop = 0}^n a_k x^k$ nor $\displaystyle g \left({x}\right) = \sum_{k \mathop = 0}^m b_k x^k$ are the null polynomial, then their leading coefficients $a_n$ and $b_m$ are non-zero.

Therefore, as $D$ is an integral domain and $a_n,b_m \in D$, so is their product $a_nb_m$.

By the definition of polynomial multiplication, it follows that $f g$ is not the null polynomial.

It follows that $D \left[{X}\right]$ has no proper zero divisors.

Hence $D \left[{X}\right]$ is an integral domain.