On formally real division algebras and quasifields of rank two
by Kalhoff, Franz
Abstract:
In view of Albert's classical result that the dimension of each ordered,
associative proper algebra over its center is necessarily infinite, it
seems not unlikely that a similar statement also holds for the rank of each
proper formally real quasifield F, i.e. for the dimension of F over its
kernel. Indeed, for some classes of ordered near fields and for ordered
quasifields admitting a real kernel, D. Gröger and J. Joussen were able to
verify that their rank must be one or infinite.
However, making use of valuation theoretical means, recently (cf. preprint
96-05 ) we were able to show that for each natural number n formally real
quasifields of rank n do exist. In this note we present a more elementary
approach to a special case of the problem in question, namely to the
existence of formally real quasifields and of formally real (unitary,
necessarily not associative) division algebras of rank two.
Key words and phrases:
1991 Mathematics Subject Classification: Primary 06F25; Secondary 17A35.
Available as
Published in: Proceedings of the 4th
International Congress of Geometry (Thessaloniki, 1996), 214--220,
Giachoudis-Giapoulis, Thessaloniki, 1997.
Contact: Franz.Kalhoff@Mathematik.Uni-Dortmund.DE