On a theorem of Frobenius (Q5894835)

The author shows that if a division ring \(D\) properly contains an algebraically closed field \(C\) of finite codimension then \(D\) is a quaternion algebra over a real-closed subfield of \(C\). As an application, it follows that the non-commutative division rings which are finite-dimensional (but not necessarily central) over the field \(\mathbb{R}\) of real numbers are quaternion algebras over real-closed fields whose algebraic closure is \(\mathbb{C}\). The author gives an example of such a division ring which is not isomorphic to the usual quaternion algebra (because its center is a real-closed field which is not isomorphic to \(\mathbb{R}\)).