On free subgroups of units in quaternion algebras

Volume 88 / 2001

Jan Krempa Colloquium Mathematicum 88 (2001), 21-27 MSC: Primary 16U60; Secondary 16H05, 11A99. DOI: 10.4064/cm88-1-3


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.


  • Jan KrempaInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland

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