Definition:Vector Space on Field Extension

Definition
Let $L/K$ be an extension of fields.

Then $L$ is a vector space over $K$.

Proof
By definition, $\left({L, +}\right)$ is an abelian group.

Moreover the laws of scalar multiplication (that is, distributivity and associativity) hold for multiplication within $L$.

Since $K \subseteq L$ these laws also hold for scalar multiplication by elements of $K$.