# Every finite extension of $\mathbb{F}_p$ is radical

Prove that if $L|\mathbb{F}_p$ is a finite extension with order $n$, then there is an $\eta\in L$ with $L=\mathbb{F}_p(\eta)$ and $\eta^{m}\in\mathbb{F}_p$ for some $m$ not divisible by $p$.

If $L$ is a finite extension of a finite field, then $L$ is a finite field itself, so has the form $L=\mathbb{F}_{q^{m}}$. Since it's an extension of $\mathbb{F}_p$, we must have $q=p$ and since $[L:\mathbb{F}_p]=n$, we have $m=n$. Now I'm left to find some $\alpha\in\mathbb{F}_{p^{n}}$ so that $\alpha^{m}\in \mathbb{F}_p$ for some $m$. But now I'm stuck, because I can't find this $\alpha$ and don't know how to prove such an $\alpha$ is a primary element of the extension $\mathbb{F}_{p^{n}}|\mathbb{F}_p$.

• Hint: the multiplicative group of $\Bbb F_{p^n}$ is cyclic. – Watson Nov 10 '16 at 16:57
• Related (or useful) : math.stackexchange.com/questions/988276 – Watson Nov 10 '16 at 17:28
• – Watson Dec 4 '16 at 13:05