Let $K$ be a field. Prove that the field of all polynomials over $K$ is a vector space over $K$.
Let $K[x]$ be the field of all polynomials over $K$. Let $\cdot$ and $+$ be the multiplication and addition defined on $K[x]$.
Let $a,b \in K[x]$ and $s \in K$. Then we can define $a +_{K[x]}b = a+b$ and
$\cdot_{K[x]}: K \times K[x] \to K[x]$ as $sa = a \cdot s(x) $ where $ K[x] \ni s(x) = 0x^n \dots0x^1 +s$
Since $K[x]$ is a field:
$$a(bc) = (ab)c) \\a(b+c)=(b+c)a = ab+ac$$
Addition is associative, has neutral element and inverse element.
Identity element of scalar multiplication: $1 = 0x^n + \dots +0x^1+1$
Is it enough or I should add something / change something?