A p-adic field is a type of number system used in mathematics, particularly in number theory. It extends the concept of the usual rational numbers by introducing a new way to measure distances based on a prime number p . In this system, numbers are represented in a way that emphasizes their divisibility by p , allowing mathematicians to study properties of numbers that are not easily visible in the standard number system. The most common example of a p-adic field is the field of p-adic numbers, denoted as \mathbbQ_p . This field includes all rational numbers, but with a different notion of convergence and limits. The p-adic numbers are useful in various areas of mathematics, including algebraic geometry and cryptography, as they provide insights into the structure of numbers and their relationships.