81 research outputs found
Signatures of hermitian forms and the Knebusch Trace Formula
Signatures of quadratic forms have been generalized to hermitian forms over
algebras with involution. In the literature this is done via Morita theory,
which causes sign ambiguities in certain cases. In this paper, a hermitian
version of the Knebusch Trace Formula is established and used as a main tool to
resolve these ambiguities.
The last page is an erratum for the published version. We inadvertently (I)
gave an incorrect definition of adjoint involutions; (II) omitted dealing with
the case . As , the
omission does not affect our reasoning or our results. For the sake of
completeness we point out where some small changes should be made in the
published version.Comment: This is the final version before publication. The last page is an
updated erratum for the published versio
Division, adjoints, and dualities of bilinear maps
The distributive property can be studied through bilinear maps and various
morphisms between these maps. The adjoint-morphisms between bilinear maps
establish a complete abelian category with projectives and admits a duality.
Thus the adjoint category is not a module category but nevertheless it is
suitably familiar. The universal properties have geometric perspectives. For
example, products are orthogonal sums. The bilinear division maps are the
simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes
the understanding that the atoms of linear geometries are algebraic objects
with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism;
hence, nonassociative division rings can be studied within this framework.
This also corrects an error in an earlier pre-print; see Remark 2.11
Pfister involutions
The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discusse
Forms in odd degree extensions and self-dual normal bases
Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe
Forms in odd degree extensions and selfdual normal bases
Introduction. Let K be a field. Springer has proved that an ani-sotropic quadratic form over K is also anisotropic over any odd degree extension of K (see [31], [14]). If the characteristic of K is not 2, this implies that two nonsingular quadratic forms that become isomorphic over an extension of odd degree of K are already isomorphic over
Algebraic lattice constellations: bounds on performance
In this work, we give a bound on performance of any full-diversity lattice constellation constructed from algebraic number fields. We show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain. Furthermore, we discuss constructions, minimum product distance, and bounds for full-diversity complex rotated Z[i]/sup n/-lattices for any dimension n, which avoid the need of component interleaving
Open Problems on Central Simple Algebras
We provide a survey of past research and a list of open problems regarding
central simple algebras and the Brauer group over a field, intended both for
experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered,
compared to v
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