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General approach of the root of a p-adic number

In this work, we applied the Newton method in the p-adic case to calculate the cubic root of a p-adic number a ∈ Q ∗ where p is a prime number, and through the calculation of the approximate solution of the equation x 3 − a = 0. We also determined the rate of convergence of this method and evaluated the number of iterations obtained in each step of the approximation.