p-adic Newton Method

Here we implement the well-known Newton method to solve an algebraic equation . The algorithm is the same as in the classical case, that is, we use the iterations

The main problem is to find an appropriate starting value since convergence might be rather slow. We tried to solve this based on the following theorem.

Let be a polynomial with p-adic integer coefficients and let there exist , such that and . Then there exists , such that and .

Here is an example.