 Solution

The extension $K/F$ is a vector space over $F$. Since $K$ is finite, a maximal linearly independent subset is finite and spans it; let its size be $d$. Relative to this basis, every element of $K$ has a unique list of $d$ coefficients in $F$. There are $q$ choices for each coefficient, soThe inclusion $F\subseteq K$ makes $d\ge 1$.