I'm studying "Note on a problem of Chowla" by Birch and Swinnerton-Dyer, it's about counting the number of values attained by a polynomial in $\mathbb{F}_p$.
In the paper, for a polinomial $f$ over $\mathbb{F}_p$ we study $N(f)$ the number of $y \in \mathbb{F}_p$ such that \begin{equation} f(x) = y \end{equation} has a root in $\mathbb{F}_p$. Without giving too much detail, $N(f)$ depends only on the degree of $f$, the Galois group of the equation over $\mathbb{F}_p(y)$ and $\mathbb{F}_p^+(y)$ where $^+$ is the algebraic closure.
The thing is I have no idea on how to study Galois groups that are not over $\mathbb{Q}$. In my case, I'm only interested this problem for cubic and quartic but if I can learn a general approach that would be great. More precisely, I would like to be able to describe the group with \begin{equation} f(x) = x^4 + ax^2 + bx \end{equation} depending on if $a$ and $b$ are $0$ or not.
For instance, the case that seems the easiest is $a=b=0$, and so I would like to study $x^4-y = 0$ which leads to the field $\mathbb{F}(y)(\zeta_4, \sqrt[4]{y})$ and i'm lost here. I don't even know how things would change whith $\mathbb{F}_p^+(y)$.
Thanks for your help !