I'm trying to understand the Euler product form for a Zeta function over a finite field. So let $k = \mathbb{F}_p$ and $X = V(\mathfrak{a})$ be an algebraic set with coordinate ring $k[X]$. I believe key point is that there is a bijection
$$ X_d / \operatorname{Gal}(\bar{k} / k) \longleftrightarrow B_d $$
between rational points of degree $d$ and maximal ideals of degree $d$. Then we claim that
$$ \# X_d = d \times \# B_d $$
I'm wondering how to show the last point, i.e. that the orbits of Galois action on $X_d$ have order precisely $d$. Does this argument work ? It descends to an action of $G = \operatorname{Gal}(k_d / k) = \langle \phi \rangle$ where $k_d = \mathbb{F}_{p^d}$, and this acts freely. Because if it didn't then for some $r < d$ we have $\phi^r(x) = x$ which implies $\phi^r(x_i) = x_i$ which implies $x_i \in \mathbb{F}_{p^r}$ has degree $r < d$. So the stabilisers $G_x$ are trivial and it follows from Orbit-Stabilizer theorem?