Ring of Polynomial Forms over Field is Vector Space

Theorem
Let $\struct {F, +, \times}$ be a field.

Let $F \sqbrk X$ be the ring of polynomials over $F$.

Then $F \sqbrk X$ is an vector space over $F$.

Proof
We already have that $F \sqbrk X$ is an integral domain.

Thus vector space axioms $V \, 0$ to $V \, 4$ are fulfilled.

Fulfilment of the remaining vector space axioms is demonstrated as follows: