Computing the Hilbert class field of real quadratic fields

Using the units appearing in Stark’s conjectures on the values of L-functions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, so that k = Q( √ dk), and let ω denote an algebraic integer such that the ring of integers of k is Ok := Z+ ωZ. An important invariant of k is its class group Clk, which is, by class field theory, associated to an Abelian extension of k, the so-called Hilbert class field, denoted by Hk. This field is characterized as the maximal Abelian extension of k which is unramified at all (finite and infinite) places. Its Galois group is isomorphic to the class group Clk; hence the degree [Hk : k] is the class number hk. There now exist very satisfactory algorithms to compute the discriminant, the ring of integers and the class group of a number field, and especially of a quadratic field (see [3] and [16]). For the computation of the Hilbert class field, however, there exists an efficient version only for complex quadratic fields, using complex multiplication (see [18]), and a general method for all number fields, using Kummer theory, which is not really satisfactory except when the ground field contains enough roots of unity (see [6], [9] or [15]). In this paper, we will explore a third way, available for totally real fields, which uses the units appearing in Stark’s conjectures [21], the so-called Stark units, to provide an efficient algorithm to compute the Hilbert class field of a real quadratic field. This method relies on the truth of Stark’s conjecture (which is not yet proved!), but still we can prove independently of the conjecture that the field obtained is indeed the Hilbert class field and thus forget about the fact that we had to use this conjecture in the first place. Of course, the possibility of using Stark units for computing Hilbert or ray class fields was known from the beginning, and was one of the motivations for Stark’s conjectures. Stark himself gave many examples. It seems, however, that a complete algorithm has not appeared in the literature, and it is the purpose of this paper to give one for the case of real quadratic fields. In Section 1, we say a few words about how to construct the Hilbert class field of k when the class number is equal to 2. Here, two methods can be used which are very efficient in this case: Kummer theory and genus field theory. In Section 2 we give a special form of Stark’s conjectures, namely the Abelian rank one conjecture applied to a particular construction. Section 3 is devoted to the description of the Received by the editor January 19, 1998 and, in revised form, September 10, 1998. 1991 Mathematics Subject Classification. Primary 11R37, 11R42; Secondary 11Y35. c ©2000 American Mathematical Society