Let the operation $\times': F \to F \sqbrk X$ be defined as follows. Let $x \in F$. Let $\mathbf y \in F \sqbrk X$ be defined as: :$\mathbf y = \displaystyle \sum_{k \mathop = 0}^n y_k X^k$ where $n = \map \deg {\mathbf y}$ denotes the degree of $\mathbf y$ Thus: :$x \times' \mathbf y := \displaystyle x \sum_{k \mathop = 0}^n y_k X^k = \displaystyle \sum_{k \mathop = 0}^n \paren {x \times y_k} X^k$ We have that $\times': F \to F \sqbrk X$ is an instance of polynomial multiplication where the multiplier $x$ is a polynomial of degree $0$. Hence, let the supposed vector space over $F$ in question be denoted in full as: :$\mathbf V = \struct {F \sqbrk X, +', \times'}_F$ where: :$+': F \sqbrk X \to F \sqbrk X$ denotes polynomial addition :$\times': F \to F \sqbrk X$ denotes the operation as defined above.  We already have that $F \sqbrk X$ is an integral domain. Thus vector space axioms $\text V 0$ to $\text V 4$ are fulfilled. By definition of $\times'$, it is seen that the remaining vector space axioms are fulfilled as follows: Let $\lambda, \mu \in F$. Let $\mathbf x, \mathbf y \in F \sqbrk X$ such that $\map \deg {\mathbf x} = m$ and $\map \deg {\mathbf y} = n$. === {{Vector-space-axiom|5}} === {{begin-eqn}} {{eqn | l = \paren {\lambda + \mu} \times' \mathbf x       | r = \paren {\lambda + \mu} \times' \sum_{k \mathop = 0}^m x_k X^k       | c =  }} {{eqn | r = \sum_{k \mathop = 0}^m \paren {\lambda + \mu} \times x_k X^k       | c = Definition of $\times'$ }} {{eqn | r = \sum_{k \mathop = 0}^m \paren {\lambda \times x_k X^k + \mu \times x_k X^k}       | c = {{Field-axiom|D}} }} {{eqn | r = \sum_{k \mathop = 0}^m \lambda \times x_k X^k + \sum_{k \mathop = 0}^m \mu \times x_k X^k       | c =  }} {{eqn | r = \lambda \times' \mathbf x + \mu \times' \mathbf x       | c = Definition of $\times'$ }} {{end-eqn}} {{qed|lemma}} === {{Vector-space-axiom|6}} === {{begin-eqn}} {{eqn | l = \lambda \times' \paren {\mathbf x + \mathbf y}       | r = \lambda \times' \paren {\sum_{j \mathop = 0}^m x_j X^j + \sum_{k \mathop = 0}^n y_k X^k}       | c =  }} {{eqn | r = \sum_{j \mathop = 0}^m \lambda \times x_j X^j + \sum_{k \mathop = 0}^n \lambda \times y_k X^k       | c = Definition of $\times'$ }} {{eqn | r = \sum_{k \mathop = 0}^{\max \set {m, n} } \paren {\lambda \times x_k + \lambda \times y_k} X^k       | c = {{Defof|Polynomial Addition}} }} {{eqn | r = \sum_{k \mathop = 0}^{\max \set {m, n} } \lambda \times \paren {x_k + y_k} X^k       | c = {{Field-axiom|D}} }} {{eqn | r = \lambda \times' \mathbf x + \lambda \times' \mathbf y       | c =  }} {{end-eqn}} {{qed|lemma}} === {{Vector-space-axiom|7}} === {{begin-eqn}} {{eqn | l = \lambda \times' \paren {\mu \times' \mathbf x}       | r = \lambda \times' \paren {\mu \times' \sum_{k \mathop = 0}^n x_k X^k}       | c =  }} {{eqn | r = \lambda \times' \paren {\sum_{k \mathop = 0}^n \mu \times x_k X^k}       | c = Definition of $\times'$ }} {{eqn | r = \sum_{k \mathop = 0}^n \lambda \times \paren {\mu \times x_k} X^k       | c = Definition of $\times'$ }} {{eqn | r = \sum_{k \mathop = 0}^n \paren {\lambda \times \mu} \times x_k X^k       | c = {{Field-axiom|M1}} }} {{eqn | r = \paren {\lambda \times \mu} \times' \sum_{k \mathop = 0}^n x_k X^k       | c = Definition of $\times'$ }} {{eqn | r = \paren {\lambda \times \mu} \times' \mathbf x       | c =  }} {{end-eqn}} {{qed|lemma}} === {{Vector-space-axiom|8}} === {{begin-eqn}} {{eqn | l = 1_F \times' \mathbf x       | r = 1_F \times' \sum_{k \mathop = 0}^n x_k X^k       | c =  }} {{eqn | r = \sum_{k \mathop = 0}^n 1_F \times x_k X^k       | c = Definition of $\times'$ }} {{eqn | r = \sum_{k \mathop = 0}^n x_k X^k       | c = {{Field-axiom|M3}} }} {{eqn | r = \mathbf x       | c =  }} {{end-eqn}} {{qed|lemma}} All vector space axioms are hence seen to be fulfilled. Hence the result. {{qed}}