Abstract/Details

## ON SOLUTIONS OF POLYNOMIAL AND MATRIX EQUATIONS OVER THE RING OF P-ADIC INTEGERS

MEAD, GEORGE FRANCIS, JR.   University of South Florida ProQuest Dissertations Publishing,  1980. 8108264.

### Abstract (summary)

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.

### Indexing (details)

Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
ON SOLUTIONS OF POLYNOMIAL AND MATRIX EQUATIONS OVER THE RING OF P-ADIC INTEGERS
Author
Number of pages
88
Degree date
1980
School code
0206
Source
DAI-B 41/10, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-204-79106-0
University/institution
University of South Florida
University location
United States -- Florida
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
8108264
ProQuest document ID
303087876