2011
DOI: 10.5539/jmr.v3n3p40
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Computation of the Cubic Root of a p-adic Number

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 $ainmathbb{Q}_{p}^{ast }$ 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 … Show more

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Cited by 8 publications
(9 citation statements)
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“…In [4], the authors gave the condition for the existence of cubic roots in Q p . We generalize the result to the q-th root, and we have the condition for the existence of a q-th root of p-adic numbers in Q p when q ≥ 2 and (p, q) = 1.…”
Section: Theorem 33 a Polynomial With Integer Coefficients Has A Romentioning
confidence: 99%
See 2 more Smart Citations
“…In [4], the authors gave the condition for the existence of cubic roots in Q p . We generalize the result to the q-th root, and we have the condition for the existence of a q-th root of p-adic numbers in Q p when q ≥ 2 and (p, q) = 1.…”
Section: Theorem 33 a Polynomial With Integer Coefficients Has A Romentioning
confidence: 99%
“…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]).…”
Section: Introductionmentioning
confidence: 99%
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“…Many efforts have been made to find solutions of p-adic equations, but it was difficult to find it out right away, so people started to research ways to compute the approximate solutions using numerical methods ( [8]). To start with, there has been a research on how to find square roots of p-adic numbers ( [10]). Also, a variety of ways to find roots has been researched, such as Newton's method or other methods in numerical analysis like secant method.…”
Section: Introductionmentioning
confidence: 99%
“…Zerzaihi, Kecies, and used a fixed point iteration to approximate the solutions of x 2 = a, a ∈ Q p in Q p . Zerzaihi and Kecies (2011) then extended the root finding problem to the cube roots in Q p of p-adic numbers by approximating the zeroes of g(x) = x 3 − a, a ∈ Q p , using the secant method.…”
Section: Introductionmentioning
confidence: 99%