Let $E\supseteq F$ be fields. If $E$ is finite, show that $E=F(u)$ for some $u\in E$.
What I've come up with so far:
Let $u\in E$ be the primitive element of $E$. Then $E^*=\langle u \rangle$. So $E=\{0\}\cup \langle u \rangle$.
I want to jump right to $E=F(u)$, but I see it is clear that $E\supseteq F(u)$, but does this justify $E\subseteq F(u)$?