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 monomial 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).

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).

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 monomial polynomials with integer coeffiecients. Let $\overline{\mathbb F}_p$ be the algebraic closure of ${\mathbb F}_p$.

Question: Is there a ``natural'' 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).

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 monomial 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).