I'm studying Chatzidakis Notes about pseudo-finite fields. During a proof she states the following (by $\mathbb{F}_p$ I mean the finite field with $p$ elements for a prime $p$):
Let $f_1(x),\ldots,f_m(x),g(x) \in \mathbb{Z}[x]$ and let $L$ be the Galois extension of $\mathbb{Q}$ obtained by adjoining all the roots of the polynomials $f_i (x)$. Assume that there is a subfield $M$ of $L$ such that $Gal(L/M)$ is cyclic and
$$M \vDash \bigwedge_{i=1} ^m \exists t f_i(t) = 0 \wedge \forall t g(t) \ne 0.$$ Then the set of prime numbers $p$ such that $\mathbb{F}_p \vDash \bigwedge_{i=1} ^m \exists t f_i(t) = 0 \wedge \forall t g(t) \ne 0$ is infinite.
She claims that the latter is a consequence of Chebotarev's Density Theorem. I understand that by a corollary of Frobenius Theorem, the number of primes $p$ such that each polynomial $f_i(x)$ has a root modulo $p$ is infinite. But what about the polynomial $g(x)$?
I don't understand how to derive the latter from Chebotarev's Theorem.