## On a theorem of Frobenius. (A propos d’un théorème de Frobenius.)(French)Zbl 1021.16007

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 $$\mathbb{R}$$).

### MSC:

 16K20 Finite-dimensional division rings 12E15 Skew fields, division rings

### Keywords:

division algebras; real-closed fields; quaternion algebras
Full Text:

### References:

 [1] Blanchard, A.. Les corps non commutatifs. P.U.F., Paris, 1972. · Zbl 0249.16001 [2] Deschamps, B.. Problèmes d’arithmétique des corps et de théorie Galois. Hermann, Paris, 1998. · Zbl 0912.12001 [3] Ribenboim, P.. L’arithmétique des corps. Hermann, Paris, 1970. · Zbl 0253.12101 [4] Serre, J.P.. Corps locaux. Hermann, Paris, 1968. · Zbl 0137.02601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.