Let $K$ be a field and $F$ be a finite extension of $K$ with degree $n$. Then can we always find a an element a in $F$ s.t $\{a^t\}_{t= 0,...,n-1}$ is a basis of $F$ ?
For example, if we had constructed $L/K$ as $K(b) = L$, then $\{b^t\}_{t=0, ..,r-1}$ where $r $ is the degree of the minimal polynomial of $b$, would form a basis for $L$;however, is this possible in general ?