Square-central elements in algebras of exponent 2
(2012)
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- (en) In this thesis, we further investigate the decomposability of exponent 2 central simple algebras over fields of characteristic different from 2. The problem will be addressed through the study of square-central elements. We first extend the notion of spaces of similitudes to the wide framework of algebras with involution, later we give a characterization of the totally decomposable involutions in terms of the existence of maximal (s, t)-families. The study of expansion properties of spaces of similitudes leads us to study conditions under which a square-central element, in an exponent 2 algebra, lies in a quaternion subalgebra. It turns out that the existence, in a 64-dimensional division algebra of exponent 2, of a square-central element which is not in a quaternion subalgebra is tied to the existence of an indecomposable algebra of exponent 2 and dimension 64. The known examples of indecomposable algebras of exponent 2 and dimension 64 are constructed over fields of cohomological dimension greater than or equal to 5. However, the existence of indecomposable algebras of exponent 2 over fields of cohomological dimension smaller than or equal to 4 was still open. As an application, we improve these examples by constructing an example of such an algebra over a field of cohomological dimension as small as possible (that is 3). Another examples are given in cohomological dimension 4 and over M(t) where M is some field of u-invariant 8 and t is an indeterminate.
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Barry, D. (2012). Square-central elements in algebras of exponent 2. https://hdl.handle.net/2078.5/162183