The Brauer–siegel Theorem

Abstract

Explicit bounds are given for the residues at s = 1 of the Dedekind zeta functions of number fields. As a consequence, a simple proof of the Brauer-Siegel theorem and explicit lower bounds for class numbers of number fields are obtained. Compared with Stark's original approach, the paper is explicit and more satisfactory for number fields containing quadratic subfields. Examples are given of fully explicit lower bounds for class numbers of various types of number fields, for example normal and non-normal number fields of odd degree, with an emphasis on cubic fields, real cyclic quartic number fields, and non-normal quartic number fields containing an imaginary quadratic subfield.

The class number one problem for some totally complex quartic number fields

“…For those two cases, we verify that f > D for d K > 57 600 and so d K d 3 (10) and (13) for d K 83 4 using (8). Since the lower bounds (11) and (13) for h K are better than (12), we conclude that (9) holds true for d K 271 4 .…”

The class-number one problem for some real cubic number fields with negative discriminants

“…In that case, using [Lou05] it should be possible to prove, as in Theorem 3, that if K is ranges over the non-cyclotomic totally complex quartic fields such that their rings of algebraic integers are of the form A K = Z[ K ], where K with | K | > 1 is the fundamental unit of K, then its class number h K goes explicitly to infinity with d K .…”

“…Bounds (4) enable us to easily list all the cubic polynomials P (X) of type (T) such that d P B.Proposition 1. (See[Lou05, Corollary 8], and compare with [Lou95, Theorem 1].) Let K be a non-normal real cubic field of negative discriminant −d K −79 507.…”

On Fields of Definition of Arithmetic Kleinian Reflection Groups II

“…While the root discriminant bounds in (4.8) represent a major step towards the enumeration of all totally real defining fields of arithmetic Kleinian reflection groups (compared for instance, with the discriminant bounds produced in [4]), they are still in general much too large for the fields to appear in existing tables of totally real number fields [29]. The primary reason that the discriminant bounds in (4.8) are as large as they are is that because we lack a good bound for h (K, 2, B) we are forced to bound h (K, 2, B) by the full class number of K. We then bound the class number of K using Louboutin's refinement of the Brauer-Siegel theorem [14]. It seems likely that these bounds are far from optimal and that every field satisfying our root discriminant bounds has 2-class number considerably smaller than the bounds produced by refinements of the Brauer-Siegel theorem.…”

“…We also present a heuristic argument which allows us to further narrow the list (Theorem 4.4). The proofs of the main theorems use Borel's volume formula [6] and results of Chinburg and Friedman [8] (as in [4]) together with a Laplace-eigenvalue bound of Luo, Rudnick and Sarnak [16], a refinement of the Odlyzko discriminant bounds that takes into account primes of small norm [25], [7] and Louboutin's improvement of the Brauer-Siegel theorem [14]. We also note that several aspects of our proof were inspired by the work of Doyle, Linowitz and Voight [9] on isospectral but not isometric arithmetic 2and 3-manifolds of small volume.…”