The main importance of p-adic numbers lies within the theory of diophantine equations. So far we have treated p-adic numbers only by means of analysis. In this chapter we will investigate p-adic numbers from the view of algebra and introduce the set of p-adic integers Zp. The central theme of this chapter is Hensel’s lemma. This theorem is very likely the most important algebraic property of the p-adic numbers. By Hensel’s lemma, in many circumstances one can decide very easily whether a polynomial has roots in Zp. It has many applications, from characterizing roots of unity to factoring polynomials.