Related papers: Isotropy of orthogonal involutions

We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can…

A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary…

In this paper, we explore the behavior of orthogonal involutions in the context of totally positive field extensions. Let $K/F$ be a totally positive extension of formally real fields. By Becher's result, if a quadratic form $q$ over $F$…

Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two…

We show that a non-hyperbolic orthogonal involution on a central simple algebra over a field of characteristic different from 2 remains non-hyperbolic over some splitting field of the algebra.

In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable, if it becomes either anisotropic or metabolic over all extensions of the…

An orthogonal involution $\sigma$ on a central simple algebra $A$, after scalar extension to the function field $\mathcal{F}(A)$ of the Severi--Brauer variety of $A$, is adjoint to a quadratic form $q_\sigma$ over $\mathcal{F}(A)$, which is…

It is shown that an anisotropic orthogonal involution in characteristic two is totally decomposable if it is totally decomposable over a separable extension of the ground field. In particular, this settles a characteristic two analogue of a…

We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or…

Given a central simple algebra with involution over an arbitrary field, \'etale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and…

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal,…

Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a…

We investigate the quadratic descent of totally decomposable algebras with involution of orthogonal type in characteristic two. Both separable and inseparable extensions are included.

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra $Q$. We classify these algebras in degree~4 and give an example of such…

We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence, we obtain that every…

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with…

We provide upper bounds on the u-invariant for skew-hermitian forms over a quaternion algebra with its canonical involution in terms of the u-invariant of the base field F of characteristic different from 2 when I^3F = 0.

Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally attaches an ideal of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of…

An algebra extension $A \| B$ is right depth two in this paper if its tensor-square is $A$-$B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $A$ with itself. For example, normal subgroups of infinite groups,…

We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric…