No abstract

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.

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.

In this paper, we compute the square roots of p-adic numbers in Qp, using the secant method. We also study the performance of this method: the speed of its convergence and the number of iterations necessary to obtain the desired precision M which represents the number of p-adic digits in the development of √ a.

This work offers an analogue of Householder's Method for solving a root-finding problem f (x) = 0 in the p-adic setting. We apply this method to calculate the square roots of a p-adic number a ∈ Qp where p is a prime number, and through the calculation of the approached solution of the p-adic polynomial equation f (x) = x 2 − a = 0. We establish the rate of convergence of this method. Finally, we also determine how many iterations are needed to obtain a specified number of correct digits in the approximate.