James Ax proved that the elementary theory of the finite fields is decidable. I found its proof on the book Fried, Jarden: "Field Arithmetic 3rd edition". Fried writes on the beginning of the chapter 6.3 that $K$ is a global field. There is a Theorem 6.3.1 b):
"Let $L/K$ be a finite Galois extension of global fields and let $\mathcal C$ be a conjugacy class in $\operatorname{Gal}(L/K)$. Then the Dirichlet's density of $\{\mathfrak p\in P(K)|\left (\frac{L/K}{\mathfrak p}\right )=\mathcal C\}$ exists and is equal to $|\mathcal C|/[L:K]$."
Then on the page 446. Jarden denotes the following:
For given sentence $\theta$ of $\mathcal{L} (\operatorname{ring})$ denote $S(\theta)=S(K,1,\theta)=\{\sigma\in \operatorname{Gal}(K)|\tilde K(\sigma)\models \theta\}$ and $A(\theta)=A(K,\theta)=\{\mathfrak p\in P(K)|\bar K_{\mathfrak p}\models \theta\}$.
Here $P(K)$ is defined on the page 115: The set of prime divisors of $K/\mathbb F_q$ is denoted by $P(K)$.
Now, the Theorem 6.3.1 is used in proving the Lemma 20.9.2 b: "Let $\lambda$ be the test sentence $$p((∃X)[f_1 (X) = 0], . . . , (∃X)[f_m (X) = 0]),$$ where $f_1 , . . . , f_m ∈ K[X]$ are separable polynomials and $p$ is a Boolean polynomial. Let $B$ be the set of all $\mathfrak p ∈ P (K)$ such that all coefficients of $f_i$ are $\mathfrak p$-integral and the leading coefficient and the discriminant of the $f_i$ ’s are $p$-units, $i = 1, . . . , m$. Denote the splitting field of $f_1 · · · f_m$ over $K$ by $L$. Let $\bar S=\{\sigma\in\operatorname{Gal}(L/K)|L(\sigma)\models \lambda\}$. Then $\sigma(A(\lambda))=|\bar S(\lambda)|/[L:K]$."
It looks like $B$ is used only in Lemma 20.9.2 a, not in 20.9.2 b.
Now, the book says that Lemma 20.9.2 b follows from the Theorem 6.3.1 b.
Why do we need the assumption that in Theorem 6.3.1 b we use global fields instead of number fields as in https://mdimeglio.github.io/papers/Chebotarevs_density_theorem.pdf Theorem 2.1?