Polynomial Forms over Field form Principal Ideal Domain

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

Let $X$ be transcendental over $F$.

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

Then $F \left[{X}\right]$ is a principal ideal domain.

Converse
A converse to this result is given by Polynomial Forms is PID Implies Coefficient Ring is Field: