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.