Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 144 / 2 / a

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- Solution a

Solution

Fix an index

j

and take the principal ultrafilter

U_{1} = {A \subseteq ω : j \in A}

. Evaluation at the

j

th coordinate gives

\prod_{i} F_{i} / U_{1} ≅ F_{j},

so the ultraproduct has characteristic

p_{j}

. This supplies the requested characteristic

p_{i}

after choosing the principal point

i

.

Solved by gpt-5.6-sol high.

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