On free subgroups of units in quaternion algebras

Jan Krempa¹

¹ Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 88 (2001) no. 1, pp. 21-27

Résumé

It is well known that for the ring ${\rm H}({\mathbb Z})$ of integral quaternions the unit group ${\rm U}( {\rm H}({\mathbb Z})$ is finite. On the other hand, for the rational quaternion algebra $ {\rm H}({\mathbb Q})$, its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ${\mathbb Z}\subset A \subseteq {\mathbb Q}$ such that the group of units ${\rm U}( {{\rm H}(A)})$ of quaternions over $A$ contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments as simple as possible.

Détail

DOI : 10.4064/cm88-1-3

Keywords: known ring mathbb integral quaternions unit group mathbb finite other rational quaternion algebra mathbb its unit group infinite even contains nontrivial subgroup note see theorem corollary intermediate rings mathbb subset subseteq mathbb group units quaternions contains nontrivial subgroup each indicate subgroup explicitly best keep arguments simple possible