Content area
Abstract
If one considers the rational numbers with the p-adic valuation, the completion is the field of p-adic numbers. The closure of the rational integers in the field of p-adic numbers is the ring of p-adic integers. It is the purpose of this paper to study the properties of the p-adic numbers and to investigate the p-acid integral solutions to certain polynomial equations whose coefficients are p-adic integers. Also investigated will be the solutions of polynomial equations over the p-adic integers where the domain is the ring of p-adic integral matrices.
The main techniques for solving these polynomial equations involve using the properties of the p-adic valuation--a non-Archimedean valuation, the properties of the field of p-adic numbers and in particular the famous Hensel's lemma together with its generalization. Hensel's lemma is sometimes called the p-adic Newton method and provides a means for generating p-adic solutions once the existence of a solution is established and an initial congruence for the solution is found modulo p. The methods of p-adic analysis illustrated in the proofs of Hensel's lemma and its generalization provide the method of solution for many of the polynomial equations considered. Also used is a p-adic version of Eisenstein's irreducibility criterion together with some theorems on power residues and finite fields.
Results and conclusions are reached as to the existence and number of p-adic integral solutions of these polynomials by examining certain conditions on the degree of the polynomial and the p-adic coefficients. Also found are additional p-adic irreducibility conditions and existence and number of p-adic integral matrix solutions to certain polynomial equations when conditions are placed on the p-adic coefficients.
