Hensel codes of square roots of p-adic numbers

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.

Halley's method for finding roots of p-adic polynomial equations

“…In 2010, for instance, Knapp and Xenophontos [11] showed how classical root-finding methods from numerical analysis can be used to calculate inverses of units modulo prime powers. This was followed, in the same year, by a work of Zerzaihi, Kecies and Knapp [16] on the application of some classical root-finding methods, such as the fixed-point method, in finding square roots of p-adic numbers through Hensel's lemma. In 2011, Zerzaihi and Kecies [17] extended the work of Zerzaihi et al [16] by computing the cubic roots of p-adic numbers via secant method.…”

“…This was followed, in the same year, by a work of Zerzaihi, Kecies and Knapp [16] on the application of some classical root-finding methods, such as the fixed-point method, in finding square roots of p-adic numbers through Hensel's lemma. In 2011, Zerzaihi and Kecies [17] extended the work of Zerzaihi et al [16] by computing the cubic roots of p-adic numbers via secant method. Following this work, Kecies and Zerzaihi considered in [10] the problem of finding the cubic roots of p-adic numbers in Q p using the Newton method.…”

ON THE EXISTENCE OF p-ADIC ROOTS

“…It is necessary to confirm the existence of the q-th root of a p-adic number in Q p before computing them( [4], [5]). There are some results of the existence of square roots of p-adic numbers and the q-th roots of unity( [1][2]).…”

General approach of the root of a p-adic number

“…The Hensel codes and their properties are studied in [2][3][4]. In [8], the authors used fixed point method to calculate the Hensel code of square root of a p-adic number a ∈ Q p , it means the first numbers of the p-adic development of the √ a. In this work, we will see how we can use classical root-finding method and explore a very interesting application of tools from numerical analysis to number theory.…”