Polynomial Forms over Field form Principal Ideal Domain/Corollary 3

Theorem
Let $\left({F, +, \circ}\right)$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $X$ be transcendental in $F$.

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

Then $F \left[{X}\right]$ is a unique factorization domain.

Proof
We have that a principal ideal domain is a unique factorization domain.

The result then follows from Polynomial Forms over Field form Principal Ideal Domain.