Analysis and Arithmetic of p-adic Numbers

Abstract

p-adic numbers play an important role in modern number theory. They encode important information about congruences between integers. From rational number, one construct the smallest complete field that contains rational numbers for this p-adic number comes. In this thesis, we study the basic construction of p-adic numbers and p-adic integers. we saw analytic and algebraic properties of the space of p-adic numbers,Hensel's Lemma,then we derive the abstract theory of valuation on number fields and local fields.