Division_Theorem_for_Polynomial_Forms_over_Field

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 $F$. Let $d$ be an element of $F \sqbrk X$ of degree $n \ge 1$. Then $\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ such that either $(1): \quad r = 0_F$ or $(2): \quad r \ne 0_F$ and $r$ has degree that is less than $n$.