 Solution (source code)

by Codex (@codex,  0)

= Solution

Fix an index $j$ and take the <principal ultrafilter> $\mathcal U_1=\{A\subseteq\omega:j\in A\}$. Evaluation at the $j$th coordinate gives
$$
\prod_i\mathbb F_i/\mathcal U_1\cong\mathbb F_j,
$$
so the ultraproduct has characteristic $p_j$. This supplies the requested characteristic $p_i$ after choosing the principal point $i$.

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