The isometry classification of hermitian forms over division algebras

Abstract

The isometry classification problem occupies a central role in the theory of quadratic and hermitian forms. This article is a survey of results on the problem for quadratic and hermitian forms over a field and also for hermitian and skew-hermitian forms over a noncommutative division algebra with involution. Rather than adopting a very abstract approach, the problems are stated in matrix or linear-algebraic terms. The known solutions depend crucially on the particular field considered, although there are some general results which are mentioned. While many of the results date back a long time, some recent results, especially those on skew-hermitian forms over a quaternion algebra over a number field, are included.