P-adic numbers and the local-to-global principle

Abstract

We introduce the notion of p-adic absolute value on the rationals. We prove up to equivalence that these and the Euclidean absolute value are the only non-trivial absolute values. Then we discuss the completion of the rationals under the p-adic absolute value to obtain p-adic numbers. Finally we discuss applications of p-adic numbers to problems in number theory, in particular we explore the local to global principle.