Free groups in quaternion algebras. (Q376041)

Let \(\bigl(\frac{a,b}{K}\bigr)=\langle 1,i,j,k\mid i^2=a,\;j^2=b,\;k=ij=-ji,\;a,b\in K\rangle\) be a generalized quaternion algebra over a field \(K\).NEWLINENEWLINE The authors prove the following result. If \(\mathfrak o_K\) is the ring of integers of the algebraic number field \(K\), then the multiplicative group generated by the units \(u=\sqrt{-1}+(i-\sqrt{-1})j\) and \(w=\sqrt{-1}+(i+\sqrt{-1})j\) in the algebra \(\bigl(\frac{-1,-1}{\mathfrak o_{\mathbb Q(\sqrt{-1})}}\bigr)\) is free.