### On the square and cube roots of p-adic numbers

#### Abstract

The study of the field of $p$-adic numbers has been an important area of research in mathematics, giving rise to several important results such as the Hasse-Minkowski Theorem and the Local-Global Principle. The analysis on the complete ultrametric space $\mathbb{Q}_p$ reveals many interesting properties that are radically different from $\mathbb{R}$, the completion of $\mathbb{Q}$ with respect to the euclidean norm. The application of different numerical methods, and the analysis of their convergence in $\mathbb{Q}_p$ has been a recent development in computational number theory. The application of the Newton-Raphson, fixed-point, and secant methods to compute for the square and cube roots of $p$-adic numbers in $\mathbb Q_p$ have been respectively addressed in \cite{ignacio, Zerzaihi2, Zerzaihi}. In this paper, we complete the problem in \cite{ignacio} by computing the $q$th root of $p$-adic numbers in $\mathbb Q_p$ where $p \leq q \leq 3$. Given a root of order $r$, we determine the order of the $n$th iterate of the Newton-Raphson method, provide sufficient conditions for its convergence, and give the number of iterations required for any desired number of correct digits in the approximate.

**How to Cite this Article:**Paul Samuel P. Ignacio, On the square and cube roots of p-adic numbers, Journal of Mathematical and Computational Science, Vol 3, No 4 (2013), 993-1003 Copyright © 2013 Paul Samuel P. Ignacio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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