 Solution

ID: past-exam-of-the-analytic-number-theory-and-elliptic-curves-course-of-the-university-of-oxford/2010/b/4/ii/solution

f^{'}

about

a_{i}

∣ f^{'} (a_{i} + h_{i}) - f^{'} (a_{i}) ∣ \leq ∣ h_{i} ∣ = d q_{i} < d .

In a Non-Archimedean absolute value, adding an error of norm strictly smaller than

d

cannot change a quantity of norm

d

. Therefore

∣ f^{'} (a_{i}) ∣ = d for every i .

This simultaneously verifies the derivative assumption used at the next step of part (i).

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