Analytic Methods

Abstract

In Chapter 2, we used Dirichlet’s theorem on arithmetical pro-gressions in order to determine the kernel Ker _{k /ℚ} α of the Artin map of a quadratic field k/ℚ (Theorem 2.21) and promised the reader its proof in Chapter 3. Therefore this is at least one raison d’être of this chapter in the book. To prove Dirichlet’s theorem he invented the L-functions. The use of Dirichlet L-functions, however, goes beyond the proof of the theorem on arithmetical progressions. It turns out that the L-functions are closely related to the law of decomposition of rational primes in algebraic number fields. The reader will learn in Chapter 4 how to use L-functions to handle class numbers of some cyclotomic and quadratic fields. For this reason, we will not rush to a proof of the theorem of arithmetical progressions. We will rather start with basic topics such as geometry of numbers due to Minkowski (1864–1909) and the famous Dirichlet’s unit theorem.