Apéritif

Abstract

The idea of considering new ways to measure the “distance” between two rational numbers, and then of considering the corresponding completions, did not arise merely from some desire to generalize, but rather from several concrete situations involving problems from algebra and number theory. The new “metrics” on ℚ will be each connected to a certain prime, and they will “codify” a great deal of arithmetic information related to that prime. The goal of this first chapter is to offer an informal introduction to these ideas. Thus, we proceed without worrying too much about mathematical rigor¹ or precision, but rather emphasizing the ideas that are behind what we are trying to accomplish. Then, in the next chapter, we will begin to develop the theory in a more formal way.