Polynomial Forms over Field form Principal Ideal Domain/Proof 2

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.

Proof
We have that Polynomial Forms over Field is Euclidean Domain.

We also have that Euclidean Domain is Principal Ideal Domain.

Hence the result.