An elementary construction of Galois quaternion extension (Q919403)

Let \(F\) be a field (characteristic \(\ne 2)\) and \(F(\sqrt{m})\) its quadratic extension. Let \(n=p^2+q^2\not\in F^2\), \(m=n+r^2\), \(mn\not\in F^2\) and \(\omega = ((mn)^{1/2}(m^{1/2}+n^{1/2})(n^{1/2}+p))^{1/2}\). It is proved that \(F(\omega\)) is a Galois quaternion extension of \(F\).