## Every element in a field extension is contained in a basis

Let $F$ be a field, $E$ a finite extension of $F$, and $\alpha \in E$ a nonzero element. Show that there exists a basis $B \subseteq E$ of $E$ over $F$ such that $\alpha \in B$.

Since $\alpha$ is nonzero, $\{\alpha\}$ is $F$-linearly independent. By this exercise, there is a basis $B \subseteq K$ over $F$ which contains $\alpha$.