p-adic Numbers

Abstract

The purpose of this chapter is to introduce the theory of p-adic numbers due to Hensel. This theory has extensive applications in number theory, algebraic geometry and algebraic functions, and is an important theory in the study of modern algebra. Before we give the rigorous definitions we give a simple introduction as to how we obtain the p-adic numbers. We recall the method of solution to the congruence

$$f(x) \equiv 0\,\,(\bmod p^1 )$$

((1))

which we discussed in chapter 2; here f(x) is a polynomial with integer coefficients and p is a prime number. Our method was first to solve the congruence

$$f(x) \equiv 0\,\,(\bmod p)$$

((2))