Abstract

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 $a\in\mathbb{Q}_{p}^{\ast }$ 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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