Polynomial Forms over Field form Principal Ideal Domain/Corollary 3

Corollary to Polynomial Forms over Field form Principal Ideal Domain
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $X$ be transcendental over $F$.

Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Then $F \sqbrk X$ is a unique factorization domain.

Proof
We have the result Principal Ideal Domain is Unique Factorization Domain.

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