I'm working my way through the exercises in a book on Galois theory. Right now I've got one exercise left in the chapter on finite fields before I continue to the next chapter. But for this one, I need clarification on what the exercise is even asking! See the title of the post for the exercise.
Is it that taking $L$ to be a splitting field for one of the polynomials, for example $x^p - 1$, gives a splitting field for each of the polynomials $x^p - a, a \in \mathbb{F}_q^*$?
To answer my question about what the exercise is asking, yes, the goal is to take $L$ to be a splitting field for any one of the polynomials and then show $L$ is, in fact, a splitting field for each of the polynomials $x^p - a, a \in \mathbb{F}_q^*$. I've reached this decision because I've been able to put together a proof. Please verify!
Write $\langle c \rangle = \mathbb{F}_q^*.$ Then $\langle c^p \rangle = \langle c \rangle = \mathbb{F}_q^*$ since $p \not\mid q - 1.$ Thus $\mathbb{F}_q^*$ contains a $p^\text{th}$ root of each element in $\mathbb{F}_q^*.$
Take $L:\mathbb{F}_q^*$ to be a splitting field for $x^p - 1.$ Then $L$ introduces all the $p^\text{th}$ roots of unity, $1, \omega_1, \dots, \omega_{p - 1}.$ But then $L$ also introduces all the $p^\text{th}$ roots of $a \in \mathbb{F}_q^*, c^k, c^k\omega_1, \dots, c^k\omega_{p - 1}$ for some $k, 1 \le k \le q - 1.$
Conversely, a splitting field for $x^p - a, a \in \mathbb{F}_q^*$ introduces all the $p^\text{th}$ roots of unity by dividing by $c^k \in \mathbb{F}_q^*.$
Therefore, $L$ is a splitting field for each $x^p - a, a \in \mathbb{F}_q^*.$