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

Dieterich, Ernst

doi:10.1090/S0002-9939-00-05445-9

Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)
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Proc. Amer. Math. Soc. 128 (2000), 3159-3166 Request permission

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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