Question on finite fields and their extensions

Question

I have been given this question in Algebra class on finite fields which I have tried to solve but to no avail, so all help appreciated. I am given $ p=13;q=p^6 $, then I am asked to prove or give a counterexample to the following claim: There exists $ \alpha \in F_q $ such that $ F_q = F_p[\alpha] $ and also $ \alpha^6 \in F_p $ I have no idea if it is true or not and really have no intuition, even after trying, as to how to approach this problem.

Any help would be appreciated thanks

Answer 1

Plan of attack:

1. Some elements of $\Bbb{F}_{13}$ are roots of unity of order twelve. Find at least one.
2. If $c$ is a root of unity of order $12$, and $\alpha^6=c$, then show that $\alpha$ is a root of unity of order $72$. Careful! It does not always hold that if $c$ is of order $n$ and $\alpha^m=c$ that $\alpha$ would be of order $mn$. But this is a special case! Why?
3. The smallest extension field of $\Bbb{F}_{13}$ that contains roots of unity of order $72$ is $\Bbb{F}_{13^m}$, where ... why don't you determine $m$!
4. Depending on the value of $m$ in step 3 you can then answer the question.

Edit: Remember that an irreducible factor of degree $k$ of $x^6-c$ has a root in $\Bbb{F}_{13^k}$.