Roots of unity in definite quaternion orders

Luis Arenas-Carmona¹

¹ Facultad de Ciencias Universidad de Chile Casilla 653, Santiago, Chile

Acta Arithmetica, Tome 170 (2015) no. 4, pp. 381-393.

Résumé

A commutative order in a quaternion algebra is called selective if it embeds into some, but not all, of the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with type number 3 or larger. The proof extends to a few other closely related orders.

Détail

DOI : 10.4064/aa170-4-5

Keywords: commutative order quaternion algebra called selective embeds maximal orders algebra known given quadratic order number field selective indefinite quaternion algebra here prove order generated cubic root unity selective definite quaternion algebra rationals type number nbsp larger proof extends few other closely related orders