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Small generators of quadratic fields and reduced elements

Authors :
Jason Lizotte
Omar Kihel
Source :
Journal of Number Theory. 132:1888-1895
Publication Year :
2012
Publisher :
Elsevier BV, 2012.

Abstract

Text Ruppert proved that there is a constant d 2 such that every imaginary quadratic number field with discriminant D K has a generator α which satisfies H ( α ) ⩽ d 2 | D K | , where H ( α ) is the height of α . The constant d 2 in Ruppertʼs result is non-effective. Ruppert conjectured that one can take d 2 = 3.2 . In the first part of this paper, we give an effective version to Ruppertʼs result and deduce Ruppertʼs conjecture in many cases. Ruppert proved some results about the height of the reduced elements in a real quadratic field. In the second part of this paper, among other results, we establish a best possible constant for a result of Ruppert connecting the heights of reduced elements and generators of quadratic fields. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ghF9_nTo3aI .

Details

ISSN :
0022314X
Volume :
132
Database :
OpenAIRE
Journal :
Journal of Number Theory
Accession number :
edsair.doi.dedup.....047b571e4454422fd44c2ecfbe0d1071
Full Text :
https://doi.org/10.1016/j.jnt.2012.02.021