A Short Review on p-Adic Numbers

Abstract

In this review work we indicated that there are two different metrics to find the distance between any two rational numbers. One of these metrics is usual absolute value |.|∞ and the other one is the p-adic absolute value |.|p , here p is a prime number. Most crucial property of this norm is that it satisfies the ultra-metric triangle inequality. In this work we gave some definitions and properties about both metric and ultra metric norms. Especially, we reviewed the construction of p-adic number field. Rational numbers have two types of completions; while one of them is real numbers field the other one is p-adic numbers field.

Keywords

• p-Adic Numbers
• Normed Fields
• Non-Archimedean Norm
• Completion of Rational Number