Small generators of quadratic fields and reduced elements

TextRuppert proved that there is a constant d2d2 such that every imaginary quadratic number field with discriminant DKDK has a generator α   which satisfies H(α)⩽d2|DK|, where H(α)H(α) is the height of α  . The constant d2d2 in Ruppertʼs result is non-effective. Ruppert conjectured that one can take d2=3.2d2=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.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ghF9_nTo3aI.