scholarly journals Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

2018 ◽  
Vol 92 (3-4) ◽  
pp. 293-315
Author(s):  
Zrinka Franusic ◽  
Borka Jadrijevic
Keyword(s):  
Quadratic Fields ◽  
Relative Power ◽  
Imaginary Quadratic Fields ◽  
Integral Bases

scholarly journals Calculating Power Integral bases by Solving Relative Thue Equations

2014 ◽  
Vol 59 (1) ◽  
pp. 79-92
Author(s):  
István Gaál ◽  
László Remete ◽  
Tímea Szabó
Keyword(s):  
Upper Bound ◽  
Efficient Algorithm ◽  
Numerical Data ◽  
Quadratic Fields ◽  
Relative Power ◽  
Imaginary Quadratic Fields ◽  
Crucial Point ◽  
Integral Bases ◽  
Thue Equation

Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

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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