This is basic ramification theory that you can find in any textbook on algebraic number theory; for instance [Neukirch]. As in my comments, I will show slightly more than nonemptiness of $\operatorname{Hom}(\overline{\mathbf Z},\overline{\mathbf F}_p)$, namely determine this set via Galois theory.

Fix an algebraic closure $\overline{\mathbf F}_p$ of $\mathbf F_p$. For a finite Galois extension $\mathbf Q \to K$ with ring of integers $\mathcal O_K$, consider the left action by precomposition
$$\operatorname{Gal}(K/\mathbf Q) \times \operatorname{Hom}_{\text{Ring}}(\mathcal O_K,\overline{\mathbf F}_p) \to \operatorname{Hom}_{\text{Ring}}(\mathcal O_K,\overline{\mathbf F}_p).$$
Now [Neukirch, Prop. I.9.1] says that the action on the set of primes $\mathfrak p \subseteq \mathcal O_K$ above $p$ is transitive (and this set is nonempty). Given such a prime $\mathfrak p$, [Neukirch, Prop. I.9.4] says that $\operatorname{Stab}_{\operatorname{Gal}(K/\mathbf Q)}(\mathfrak p) \to \operatorname{Gal}((\mathcal O_K/\mathfrak p)/\mathbf F_p)$ is surjective. But $\operatorname{Hom}(\mathcal O_K,\overline{\mathbf F}_p)$ is a torsor under $\operatorname{Gal}((\mathcal O_k/\mathfrak p)/\mathbf F_p)$ by Galois theory, so the action above is transitive. The stabiliser of $\phi \colon \mathcal O_K \to \overline{\mathbf F}_p$ is the inertia subgroup of $\ker \phi$, more or less by definition [Neukirch, Prop. I.9.6].

Now we just need a straightforward limit argument. Recall that $\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)$ is the limit of $\operatorname{Gal}(K/\mathbf Q)$ for all finite Galois extensions $\mathbf Q \to K$ (with its natural profinite topology), and likewise $\operatorname{Hom}(\overline{\mathbf Z},\overline{\mathbf F}_p)$ is the limit of $\operatorname{Gal}(\mathcal O_K,\overline{\mathbf F}_p)$ over all finite Galois extensions $\mathbf Q \to K$. Since a cofiltered limit of finite nonempty sets is nonempty [Tag 0A2R] (or a countable cofiltered limit of surjective maps is nonempty), we see $\operatorname{Hom}(\overline{\mathbf Z},\overline{\mathbf F}_p) \neq \varnothing$, and a limit of torsors under finite groups is a torsor under the limit group.

**References.**

[Neukirch] J. Neukirch, *Algebraic number theory*. Grundlehren der Mathematischen Wissenschaften **322**. Springer, 1999. ZBL0956.11021.

thealgebraic closure of $\mathbf F_p$ (ortheset of algebraic integers) doesn't make sense; it is only defined up to non-canonical isomorphism. $\endgroup$`\mathbb`

takes a following argument (unlike, e.g.,`\bf`

), so, e.g.,`{\mathbb F}`

is the same as`{\mathbb{F}}`

, and you might as well drop the outer braces for`\mathbb{F}`

(or`\mathbb F`

). $\endgroup$1more comment