Complex Multiplication

Abstract

In this chapter we want to describe how points of finite order on certain elliptic curves can be used to generate interesting extension fields of ℚ. Here we mean points of finite order with arbitrary complex coordinates, not just the ones with rational coordinates that we studied in Chapter II. So we will need to use some basic theorems about extension fields and Galois theory, but nothing very fancy. We will start by reminding you of most of the facts we will be using, and you can look at any basic algebra text (such as Herstein [1] or Jacobson [1]) for the proofs and additional background material.