Past exam of the options II course of the University of Oxford / 2009 / A / A2 / b

Regard $K$ as a vector space over its prime subfield ${\mathbb{F}}_p$. Since $K$ is finite, a maximal linearly independent list terminates and is a finite basis, of some dimension $n\ge 1$. Each field element has a unique coordinate vector in ${\mathbb{F}}_p^n$, so