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: | , , |
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| Format: | text |
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Archīum Ateneo
2016
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| Subjects: | |
| 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 |
| Summary: | 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. |
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