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Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)


Author: Ernst Dieterich
Journal: Proc. Amer. Math. Soc. 128 (2000), 3159-3166
MSC (2000): Primary 17A35, 17A45, 57S25
DOI: https://doi.org/10.1090/S0002-9939-00-05445-9
Published electronically: May 18, 2000
MathSciNet review: 1690982
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Abstract:

In his remarkable article ``Quadratic division algebras'' (Trans. Amer. Math. Soc. 105 (1962), 202-221), J. M. Osborn claims to solve `the problem of determining all quadratic division algebras of order 4 over an arbitrary field $F$ of characteristic not two $\ldots$ modulo the theory of quadratic forms over $F$' (cf. p. 206). While we shall explain in which respect he has not achieved this goal, we shall on the other hand complete Osborn's basic results (by a reasoning which is finer than his) to derive in the real ground field case a classification of all 4-dimensional quadratic division algebras and the construction of a 49-parameter family of pairwise nonisomorphic 8-dimensional quadratic division algebras.

To make these points clear, we begin by reformulating Osborn's fundamental observations on quadratic algebras in categorical terms.


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Additional Information

Ernst Dieterich
Affiliation: Uppsala Universitet, Matematiska Institutionen, Box 480, S-751 06 Uppsala, Sverige
Email: Ernst.Dieterich@math.uu.se

DOI: https://doi.org/10.1090/S0002-9939-00-05445-9
Received by editor(s): December 8, 1998
Received by editor(s) in revised form: January 4, 1999
Published electronically: May 18, 2000
Communicated by: Lance W. Small
Article copyright: © Copyright 2000 American Mathematical Society