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Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)
Author(s):
Ernst
Dieterich
Journal:
Proc. Amer. Math. Soc.
128
(2000),
3159-3166.
MSC (2000):
Primary 17A35, 17A45, 57S25
Posted:
May 18, 2000
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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  of characteristic not two  modulo the theory of quadratic forms over  ' (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:
10.1090/S0002-9939-00-05445-9
PII:
S 0002-9939(00)05445-9
Received by editor(s):
December 8, 1998
Received by editor(s) in revised form:
January 4, 1999
Posted:
May 18, 2000
Communicated by:
Lance W. Small
Copyright of article:
Copyright
2000,
American Mathematical Society
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