Let $p$ be a prime and ${\mathbb F}\_p$ the finite field with $p$ elements. There is a canonical ring map ${\mathbb Z} \to {\mathbb F}\_p \cong {\mathbb Z}/ p {\mathbb Z}$. Denote the image of $n$ by $[n]\_p$.


Now consider the set of algebraic integers $\overline {\mathbb Z} \subset {\mathbb C}$, which is the set of roots of monic polynomials with integer coeffiecients. Let $\overline{\mathbb F}\_p$ be the algebraic closure of ${\mathbb F}\_p$.


**Question:** Is there a ring map $\overline{\mathbb Z} \to \overline{\mathbb F}\_p$ which extends the natural map ${\mathbb Z} \to {\mathbb F}\_p$? For example, such a map should send $a + b \sqrt{2}$ to $[a]\_p + [b]\_p \sqrt{2}$ (if this ever makes sense).



A ring map from algebraic integers to algebraic closure of $\mathbb F_p$
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](https://stacks.math.columbia.edu/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*](https://doi.org/10.1007/978-3-662-03983-0). Grundlehren der Mathematischen Wissenschaften **322**. Springer, 1999. [ZBL0956.11021](https://zbmath.org/?q=an:0956.11021).


