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The Brauer–Siegel Theorem
Author(s) -
Louboutin Stéphane R.
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705006654
Subject(s) - mathematics , quartic function , algebraic number field , quadratic equation , dedekind cut , class number , pure mathematics , class (philosophy) , dedekind sum , discriminant , number theory , field (mathematics) , brauer group , discrete mathematics , geometry , artificial intelligence , computer science
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.