In this work, we applied the classical numerical method of the secant in the p-adic case to calculate the cubic root of a p-adic number $ainmathbb{Q}_{p}^{ast }$ where $p$ is a prime number, and this 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.

Computing both the cubic root and other roots of a p-adic number is useful both for their theoretical values as for their theoretical applications in the field of theoretical computer science and cryptography.

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In this work we are concerned with the calculation of the Hensel codes of square roots of p-adic numbers, using the fixed point method and this through the calculation of the approached solution of f (x) = x 2 − a = 0 in Qp. We also determine the speed of convergence and the number of iterations.

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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 * p 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.