Polynomial Forms over Field form Principal Ideal Domain/Corollary 1
Let $X$ be transcendental over $F$.
Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.
Let $f$ be an irreducible element of $F \sqbrk X$.
It follows from Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal that $\ideal f$ is maximal for irreducible $f$.