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

Deschamps, Bruno

Ann. Math. Blaise Pascal 8, No. 2, 61-66 (2001).

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 \(\mathbb{R}\)).

Reviewer: J.-P.Tignol (Louvain-La-Neuve)

MSC:

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

Keywords:

division algebras; real-closed fields; quaternion algebras

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