Ahmet Tekcan, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: tekcan@uludag.edu.tr

Arzu Ozkoc, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: aozkoc@uludag.edu.tr

Hatice Alkan, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: halkan@uludag.edu.tr

Abstract/Résumé:

Let \(k\geq 2\) be an integer and let \(D = k^2+k+1\) be a positive non-square integer. In this work, we derive some properties (including cycles) of ideals \(I_1 = [k,k-1+\sqrt{D}]\), \(I_2 = [k+1,k+\sqrt{D}]\) and their product \(I\). In the last section, we consider the indefinite binary quadratic forms \(F_{I_1}\), \(F_{I_2}\) and \(F_I\) of discriminant \(\Delta=4D\) which correspond to \(I_1\), \(I_2\) and \(I\), respectively and we formulate the cycle of \(F_{I_1}\) and \(F_{I_2}\).

Soit \(k\ge 2\) un entier tel que \(D = k^2+k+1\) ne soit pas le carré d’un entier. Dans ce travail, on obtient quelques propriétés (incluant des cycles) des idéaux \(I_1 = [k,k-1+\sqrt{D}]\), \(I_2 = [k+1,k+\sqrt{D}]\) et de leur produit \(I\). Dans le dernier paragraphe, on considère les formes quadratiques binaires indéfinies \(F_{I_1}\), \(F_{I_2}\) et \(F_I\) de discriminant \(\Delta=4D\) qui correspondent respectivement aux idéaux \(I_1\), \(I_2\) et \(I\) à fin de formuler le cycle de \(F_{I_1}\) et \(F_{I_2}\).

Keywords: cycles of ideals, cycles of quadratic forms, ideals, product ideals, quadratic forms

AMS Subject Classification: Quadratic forms over general fields 11E04

PDF(click to download): On Cycles and Products of Ideals and Corresponding Indefinite Quadratic Forms