The Class Number One Problem

Abstract

In Gauss' book Disquisitiones Arithmeticae, he proposed the following three problems: (i) The class number h(D) ! 1 as D ! 1: (ii) For a given class number n, giving a complete list of imaginary quadratic elds of class number n: (iii) There are in nitely many real quadratic elds of class number one. The rst problem was proved by Heilbronn in 1934. The third one, however, is still unsolved, more in- formation can be found in Mollin [13]. In this thesis, we only concern ourselves with the second problem, which is known as the class number problem. More precisely, we will only focus on the special case n = 1: For n = 1; we call this problem the class number one problem, and it has an interesting history. In the book Disquisitiones Arithmeticae, Gauss gave nine imaginary quadratic elds of class number one, and he conjectured these are all imaginary quadratic elds of class number one. It turns out his answer was correct. However, the rst proof of class number one problem was published in 1952. In 1952, Heegner [8] gave a proof, but the proof was not accepted at the rst because relied on Weber. Then Baker [1] in 1966, and Stark [17] in 1967 independently gave a complete correct proof. Later, in 1969, Stark [18] lled the gap in the Heegner's proof. For other values of n; byWatkins' work [20] in 2003, the class number problem has been solved for all n 100: This thesis will cover quadratic forms, number elds (especially orders and ideals in quadratic elds), mod- ular functions and class eld theory. Then, by combining results from these areas, we will give a proof of the class number one problem. In the rst chapter, we will talk about quadratic forms, and prove the niteness of the order of the form class group C(D); then state the class number one problem in terms of this group. The second chapter will focus on number elds. In this chapter, we will introduce the Artin symbol and the Artin map. Then we will study ideals and orders in imaginary quadratic elds, and construct the connection between the form class group and the ideal class group. At the end of this chapter, we will introduce the ring class eld and then prove theorems and properties by using previous results. In chapter three, our goal is to construct the ring class eld that we introduced in chapter two. We will begin with modular functions, especially the j-invariant, and then introduce the modular equation. In the section 3.7, we will present the main theorem of this chapter, and the next few sections we will talk about the function 2(z) and the Weber functions that can also generate the ring class elds. In the last chapter, we will give the proof of the class number one problem.