Los Angeles Math Circle

4/8 -- LAMC High School: Absolute Values and the p-adics -- Jaclyn Lang

We are all familiar with the absolute value function. It measures the distance of a number from 0. We will examine the essential properties of this function, define other absolute values on the rational numbers, and classify all such absolute values. This leads to the idea of completions and the p-adic numbers which we discuss briefly. Finally we introduce Hensel's Lemma for solving polynomial equations over the p-adics and mention the local-global philosophy.