The Character Group of Q

The characters of a finite abelian group G are the homomorphisms from G to the unit circle S1 = {z ∈ C : |z| = 1}. Two characters can be multiplied pointwise to define a new character, and under this operation the set of characters of G forms an abelian group, with identity element the trivial character, which sends each g ∈ G to 1. Characters of finite abelian groups are important, for example, as a tool in estimating the number of solutions to equations over finite fields. [3, Chapters 8, 10]. The extension of the notion of a character to nonabelian or infinite groups is essential to many areas of mathematics, in the context of harmonic analysis or representation theory, but here we will focus on discussing characters on one of the simplest infinite abelian groups, the rational numbers Q. This is a special case of a situation that is well-known in algebraic number theory, but all references I could find in the literature are based on [4], where one can’t readily isolate the examination of the character group of Q without assuming algebraic number theory and Fourier analysis on locally compact abelian groups. (In [2, Chapter 3,§1], the determination of the characters of Q is made without algebraic number theory, but the Pontryagin duality theorem from Fourier analysis is used at the end.) The prerequisites for the discussion here are more elementary: familiarity with the complex exponential function, the p-adic numbers Qp, and a few facts about abelian groups. In particular, this discussion should be suitable for someone who has just learned about the p-adic numbers and wants to see how they can arise in answering a basic type of question about the rational numbers. Concerning notation, r and s will denote rational numbers, p and q will denote prime numbers, and x and y will denote real or p-adic numbers (depending on the context). The word “homomorphism” will always mean “group homomorphism”, although sometimes we will add the word “group” for emphasis. The sets Q, R, Qp, and the p-adic integers Zp, will be regarded primarily as additive groups, with multiplication on these sets being used as a tool in the study of the additive structure.