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 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.
Ignacio, P.S., Addawe, J., Nable, J. (2016/08). P-adic Qth roots via newton-raphson method. Thai Journal of Mathematics, 14(2), 417-429.