scholarly journals On $2$-class field towers of imaginary quadratic number fields

1994 ◽  
Vol 6 (2) ◽  
pp. 261-272 ◽  
Author(s):  
Franz Lemmermeyer
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Class Field ◽  
Quadratic Number Fields ◽  
Class Field Towers ◽  
Imaginary Quadratic Number

scholarly journals Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields: Corrigenda

1994 ◽  
Vol 50 (2) ◽  
pp. 351-352
Author(s):  
Elliot Benjamin
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Class Field ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Hilbert Class Field

In my earlier paper [1] I made the claim that there are three groups in Hall and Senior's book “The Groups of Order 2n(n ≤ 6)” that are in error (groups 64/140, 64/141, 64/143). However, it has been pointed out to me by Franz Lemermeyer that I made the unfortunate oversight of using the definition [x, y] = xyx−1y−1 for the commutator whereas Hall and Senion use the definition [x, y] = x−1y−1xy (see [2]). With this correction there is no problem with the above three groups in Hall and Senior.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

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