Quadratic descent of generalized quadratic forms over a division algebra with involution of the first kind in characteristic two is investigated. Using the notion of transfer, it is shown that …
Quadratic descent of generalized quadratic forms over a division algebra with involution of the first kind in characteristic two is investigated. Using the notion of transfer, it is shown that a system of quadratic forms, associated to such a generalized quadratic form, can be used to characterize its descent properties.
It is shown that an anisotropic [Formula: see text]-fold system of hermitian forms over a quaternion algebra in characteristic two remains anisotropic over all odd degree extensions of the base …
It is shown that an anisotropic [Formula: see text]-fold system of hermitian forms over a quaternion algebra in characteristic two remains anisotropic over all odd degree extensions of the base field. It is also shown that an anisotropic orthogonal involution on a central simple algebra of co-index 3 in characteristic two is anisotropic over any odd degree extension of the ground field.
Abstract There exists a mistake in the proof of Theorem 4.2. We present a new proof of this theorem, which shows that the main results of the paper are still …
Abstract There exists a mistake in the proof of Theorem 4.2. We present a new proof of this theorem, which shows that the main results of the paper are still true.
A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this …
A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.
In characteristic two, we determine all possible total decompositions of a totally decomposable algebra with orthogonal involution into tensor products of quaternion algebras with involution.
In characteristic two, we determine all possible total decompositions of a totally decomposable algebra with orthogonal involution into tensor products of quaternion algebras with involution.
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of …
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.
We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension …
We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.
We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension …
We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.
Some necessary and sufficient conditions are obtained for a totally decomposable algebra with orthogonal involution in characteristic two to have a separable descent.
Some necessary and sufficient conditions are obtained for a totally decomposable algebra with orthogonal involution in characteristic two to have a separable descent.
The algebra of similitudes of totally singular generalized quadratic forms in characteristic two is investigated. It is shown that this algebra satisfies certain functorial properties. An application of this study …
The algebra of similitudes of totally singular generalized quadratic forms in characteristic two is investigated. It is shown that this algebra satisfies certain functorial properties. An application of this study to central simple algebras with orthogonal involutions is also given.
We investigate the Pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their Pfaffians and some other related …
We investigate the Pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their Pfaffians and some other related invariants is studied. Also, in arbitrary characteristic, a criterion is obtained for an orthogonal involution on a biquaternion algebra to be metabolic.
A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only …
A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.
We associate to every symmetric (antisymmetric) Hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even …
We associate to every symmetric (antisymmetric) Hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even Hermitian forms are isometric if and only if their associated systems are equivalent. As an application, it is also shown that an anisotropic symmetric Hermitian form over a quaternion division algebra in characteristic two remains anisotropic over all odd degree extensions of the ground field.
In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.
In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.
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 …
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 conjecture formulated by Bayer-Fluckiger et al.
We use the pfaffian to study some descent properties of biquaternion algebras with involution of the first kind in arbitrary characteristic.
We use the pfaffian to study some descent properties of biquaternion algebras with involution of the first kind in arbitrary characteristic.
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 is adjoint …
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 is adjoint to a bilinear Pfister form over all splitting fields of the algebra. A stronger result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra.
We obtain some criteria for a symmetric square-central element of a totally decomposable algebra with orthogonal involution in characteristic two, to be contained in an invariant quaternion subalgebra.
We obtain some criteria for a symmetric square-central element of a totally decomposable algebra with orthogonal involution in characteristic two, to be contained in an invariant quaternion subalgebra.
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 …
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 be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution.
We obtain some criteria for a symmetric square-central element of a totally decomposable algebra with orthogonal involution in characteristic two, to be contained in an invariant quaternion subalgebra.
We obtain some criteria for a symmetric square-central element of a totally decomposable algebra with orthogonal involution in characteristic two, to be contained in an invariant quaternion subalgebra.
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 …
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 conjecture formulated by Bayer-Fluckiger et al.
We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the …
We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the Wall form and some invariants of the induced involution on the Clifford algebra is studied.
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 investigate the quadratic descent of totally decomposable algebras with involution of orthogonal type in characteristic two. Both separable and inseparable extensions are included.
We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the …
We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the Wall form and some invariants of the induced involution on the Clifford algebra is studied.
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is …
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field.
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is …
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field.
We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the …
We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the Wall form and some invariants of the induced involution on the Clifford algebra is studied.
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is …
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field.
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 investigate the quadratic descent of totally decomposable algebras with involution of orthogonal type in characteristic two. Both separable and inseparable extensions are included.
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians is studied. Also, in …
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians is studied. Also, in arbitrary characteristic, a criterion for an orthogonal involution on a biquaternion algebra to be metabolic is obtained.
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians and some other related …
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians and some other related invariants is studied. Also, in arbitrary characteristic, a criterion is obtained for an orthogonal involution on a biquaternion algebra to be metabolic.
Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, …
Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences, and their decompositions, are investigated.
A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, …
A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type.
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras …
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras …
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.
This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups. It aims to provide the algebra-theoretic foundations for much …
This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups. It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields. Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms. Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras. Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4. For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras.
Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions …
Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.
If <italic>K/F</italic> is a quadratic extension, we give necessary and sufficient conditions in terms of the discriminant (resp. the Clifford algebra) for a quadratic form of dimension 2 (resp. 4) …
If <italic>K/F</italic> is a quadratic extension, we give necessary and sufficient conditions in terms of the discriminant (resp. the Clifford algebra) for a quadratic form of dimension 2 (resp. 4) over <italic>K</italic> to be similar to a form over <italic>F</italic>. We give similar criteria for an orthogonal involution over a central simple algebra <italic>A</italic> of degree 2 (resp. 4) over <italic>K</italic> to be such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper A prime circled-times Subscript upper F Baseline upper K"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo>⊗</mml:mo> <mml:mi>F</mml:mi> </mml:msub> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A = A’ { \otimes _F}K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A prime"> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">A’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is invariant under the involution. This leads us to an example of a quadratic form over <italic>K</italic> which is not similar to a form over <italic>F</italic> but such that the corresponding involution comes from an involution defined over <italic>F</italic>.
ABSTRACT Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. …
ABSTRACT Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ϕ ∈ I 2(F) such that dim ϕ = 8, [C(ϕ)] = [D], and ϕ does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M 2 m (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 n−2 − 2 and n ≥ 5, we show that q σ ∈ I n (K), where q σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.
Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, …
Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences, and their decompositions, are investigated.
We associate to every symmetric (antisymmetric) Hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even …
We associate to every symmetric (antisymmetric) Hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even Hermitian forms are isometric if and only if their associated systems are equivalent. As an application, it is also shown that an anisotropic symmetric Hermitian form over a quaternion division algebra in characteristic two remains anisotropic over all odd degree extensions of the ground field.
The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers …
The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Jacobs University, Bremen, GermanyKatrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)
We define an invariant of torsors under adjoint linear algebraic groups of type C n -equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that …
We define an invariant of torsors under adjoint linear algebraic groups of type C n -equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H 3 (F, µ 2 ).The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups.We also prove that the invariant detects whether a central simple algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.
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 …
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 quadratic forms q, q^\prime over F , if q_L \cong q^\prime_L then q \cong q^\prime . It is natural to ask whether similar results hold for algebras with involution. We give a general overview of recent and important progress on these three questions, with particular attention to the relevance of hyperbolicity, isotropy and isomorphism over some appropriate function field. In addition, we prove the anisotropy property in some new low degree cases.
In this note we effect a reduction of the theory of hermitian forms of two particular types (coefficients in a quadratic field or in a quaternion algebra with the usual …
In this note we effect a reduction of the theory of hermitian forms of two particular types (coefficients in a quadratic field or in a quaternion algebra with the usual anti-automorphism) to that of quadratic forms.The main theorem ( §2) enables us to apply directly the known results on quadratic forms.This is illustrated in the discussion in §3 of a number of special cases.Let <£ be an arbitrary quasi-field of characteristic different from 2 in which an involutorial anti-automorphism a-^â is defined.For the present we do not exclude the cases where <£ is commutative and a=a or $ is a quadratic field with a-»S as its automorphism.Suppose 9Î is an ^-dimensional vector space over $.We define a bilinear form (x, y) as a function of pairs of vectors with values in $, such that
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 is adjoint …
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 is adjoint to a bilinear Pfister form over all splitting fields of the algebra. A stronger result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra.
Let A be a central simple algebra over a field F. Let k1,…, kr be cyclic extensions of F such that k1 ⊗F… ⊗Fkr is a field. We investigate conditions …
Let A be a central simple algebra over a field F. Let k1,…, kr be cyclic extensions of F such that k1 ⊗F… ⊗Fkr is a field. We investigate conditions under which A is a tensor product of symbol algebras where each field ki lies in a symbol F-algebra factor of the same degree as ki over F. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians is studied. Also, in …
We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians is studied. Also, in arbitrary characteristic, a criterion for an orthogonal involution on a biquaternion algebra to be metabolic is obtained.
Given a regular symmetric and isotropic bilinear form on a finite-dimensional vector space V (dim(V) ≥ 3) over a commutative field K of characteristic ≠ 2. The commutatorgroup Ω(V) of …
Given a regular symmetric and isotropic bilinear form on a finite-dimensional vector space V (dim(V) ≥ 3) over a commutative field K of characteristic ≠ 2. The commutatorgroup Ω(V) of the orthogonal group O(V) is generated by the set of Eichler-transformations. We will compute for each π ∈ Ω(V) the length 1 (π), i. e. the minimal number of Eichler-transformations in a product that equals π.
We study Pfister neighbors and their characterization over fields of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where we include the case of singular …
We study Pfister neighbors and their characterization over fields of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>φ</mml:mi> <mml:annotation encoding="application/x-tex">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is dominated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. From this, we derive an analogue in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a result by Knebusch saying that, in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="not-equals 2"> <mml:semantics> <mml:mrow> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\neq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch’s result generally fails in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for singular forms. As an application, we characterize certain forms of height <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.
We will assume throughout thatFis a field of characteristic char F≠2 and thatVis a non-degenerate quadratic space overFof finite dimension dim V=n. The orthogonal group ofVis denotedOn(V) and Ωn(V) is …
We will assume throughout thatFis a field of characteristic char F≠2 and thatVis a non-degenerate quadratic space overFof finite dimension dim V=n. The orthogonal group ofVis denotedOn(V) and Ωn(V) is its commutator subgroup. John Hsia, in reference to the classical fact that every element of Ωn(V) is a product of commutators of symmetries, asked the following very basic question: ForFa local field, does there exist a boundk, depending only onn, such that every element of Ωn(V) is a product ofksuch commutators or fewer? A corollary of the results of this article answers this question completely for a non-dyadicF: Every element in Ωn(V) is a product of [n/2] such commutators, except for a “handful” of elements which can be listed (all are certain types of involutions whenn≥6), where [n/2]+1 factors are required. (See Theorem 4.) Of course, Hsia's question can be asked for anyF, and most of the analysis is carried out in the more general context.
A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, …
A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type.