### P-Adic Qth Roots Via Newton-Raphson Method

#### Abstract

Henselâ€™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.

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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