Polynomial Forms over Field form Principal Ideal Domain/Corollary 1

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 $F$.

If $f$ is an irreducible element of $F \left[{X}\right]$, then $F \left[{X}\right] / \left({f}\right)$ is a field.

Proof
It follows from Principal Ideal of Irreducible Element that $\left({f}\right)$ is maximal for irreducible $f$.

Therefore by Maximal Ideal iff Quotient Ring is Field, $F \left[{X}\right] / \left({f}\right)$ is a field.