P-Adic Qth Roots Via Newton-Raphson Method

Hensel’s lemma has been the basis for the computation of the square roots of p-adic numbers in Zp. We generalize this problem to the computation of qth roots of p-adic numbers in Qp, where q is a prime and p is greater than q. We provide necessary and sufficient conditions for the existence of qth r...

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Main Authors:	Ignacio, Paul Samuel, Addawe, Joel, Nable, Job A
Format:	text
Published:	Archīum Ateneo 2016
Subjects:	p-adic number Newton-Raphson p-adic roots Analysis Mathematics
Online Access:	https://archium.ateneo.edu/mathematics-faculty-pubs/10 http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/937
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Institution:	Ateneo De Manila University

id	ph-ateneo-arc.mathematics-faculty-pubs-1009
record_format	eprints
spelling	ph-ateneo-arc.mathematics-faculty-pubs-10092020-02-27T07:45:52Z P-Adic Qth Roots Via Newton-Raphson Method Ignacio, Paul Samuel Addawe, Joel Nable, Job A Hensel’s lemma has been the basis for the computation of the square roots of p-adic numbers in Zp. We generalize this problem to the computation of qth roots of p-adic numbers in Qp, where q is a prime and p is greater than q. We provide necessary and sufficient conditions for the existence of qth roots of p-adic numbers in Qp. Then, given a root of order r, we use the Newton-Raphson method to approximate the qth root of a p-adic number a. We also determine the rate of convergence of this method and the number of iterations needed for a specified number of correct digits in the approximate. 2016-01-01T08:00:00Z text https://archium.ateneo.edu/mathematics-faculty-pubs/10 http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/937 Mathematics Faculty Publications Archīum Ateneo p-adic number Newton-Raphson p-adic roots Analysis Mathematics
institution	Ateneo De Manila University
building	Ateneo De Manila University Library
country	Philippines
collection	archium.Ateneo Institutional Repository
topic	p-adic number Newton-Raphson p-adic roots Analysis Mathematics
spellingShingle	p-adic number Newton-Raphson p-adic roots Analysis Mathematics Ignacio, Paul Samuel Addawe, Joel Nable, Job A P-Adic Qth Roots Via Newton-Raphson Method
description	Hensel’s lemma has been the basis for the computation of the square roots of p-adic numbers in Zp. We generalize this problem to the computation of qth roots of p-adic numbers in Qp, where q is a prime and p is greater than q. We provide necessary and sufficient conditions for the existence of qth roots of p-adic numbers in Qp. Then, given a root of order r, we use the Newton-Raphson method to approximate the qth root of a p-adic number a. We also determine the rate of convergence of this method and the number of iterations needed for a specified number of correct digits in the approximate.
format	text
author	Ignacio, Paul Samuel Addawe, Joel Nable, Job A
author_facet	Ignacio, Paul Samuel Addawe, Joel Nable, Job A
author_sort	Ignacio, Paul Samuel
title	P-Adic Qth Roots Via Newton-Raphson Method
title_short	P-Adic Qth Roots Via Newton-Raphson Method
title_full	P-Adic Qth Roots Via Newton-Raphson Method
title_fullStr	P-Adic Qth Roots Via Newton-Raphson Method
title_full_unstemmed	P-Adic Qth Roots Via Newton-Raphson Method
title_sort	p-adic qth roots via newton-raphson method
publisher	Archīum Ateneo
publishDate	2016
url	https://archium.ateneo.edu/mathematics-faculty-pubs/10 http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/937
_version_	1681506522314571776

P-Adic Qth Roots Via Newton-Raphson Method

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