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Journal of the London Mathematical Society 2005 72(1):40-52; doi:10.1112/S0024610705006654
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© The London Mathematical Society

The Brauer–Siegel Theorem

Stéphane R. Louboutin

Institut de Mathématiques de Luminy UMR 6206, 163 Avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France loubouti{at}iml.univ-mrs.fr

Received 21 January 2004. Revision received 22 September 2004.

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.


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