Orthogonal Pfister involutions in characteristic two

Dolphin, Andrew E.

doi:10.1016/j.jpaa.2014.02.013

Cited by 23 publications

(49 citation statements)

References 10 publications

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“…Now, let

α_{i} \in F^{\times}

be a representative of the class

disc σ_{i} \in F^{\times} / F^{\times 2}

i = 1, \dots, n

. By [, (7.3)], the bilinear Pfister form

⟨ false⟨ α_{1}, \dots, α_{n} false⟩ ⟩

is independent of the decomposition of

(A, σ)

. As in , we denote this form by

Pf (A, σ)

.…”

Section: Orthogonal Pfister Involutinsmentioning

confidence: 99%

“…Since an orthogonal involution in characteristic two cannot be hyperbolic, one may replace the hyperbolicity condition with metabolicity. Also, using [, (5.5)] one can find a metabolic bilinear form

frakturb

of dimension

2^{n}

over a field

F

which is not similar to any Pfister form. This can be used to show that the implication

(3) \Rightarrow (2)

is not true in characteristic two.…”

Section: Introductionmentioning

confidence: 99%

“…However, by [, (5.5)] an anisotropic symmetric bilinear form of dimension

2^{n}

over

F

, which is either anisotropic or metabolic over all extensions of

F

, is similar to a Pfister form. Considering this result a conjecture may be formulated as follows (see ): Conjecture Let

(A, σ)

be a central simple algebra of degree

2^{n}

with orthogonal involution over a field

F

of characteristic two. If

σ

is anisotropic, then the following statements are equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…The implication

(1) \Rightarrow (2)

follows from [, (6.2)] (even without the anisotropy condition on

σ

). The implication

(2) \Rightarrow (3)

was also proved in [, (8.3)]. The aim of this work is to prove the implication

(3) \Rightarrow (1)

.…”

Section: Introductionmentioning

confidence: 99%

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Pfister involutions in characteristic two

Nokhodkar

2017

Bull. London Math. Soc.

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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 ground field. A similar result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra and it becomes adjoint to a bilinear Pfister form over all splitting fields of the algebra.

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“…Now, let

α_{i} \in F^{\times}

be a representative of the class

disc σ_{i} \in F^{\times} / F^{\times 2}

i = 1, \dots, n

. By [, (7.3)], the bilinear Pfister form

⟨ false⟨ α_{1}, \dots, α_{n} false⟩ ⟩

is independent of the decomposition of

(A, σ)

. As in , we denote this form by

Pf (A, σ)

.…”

Section: Orthogonal Pfister Involutinsmentioning

confidence: 99%

frakturb

of dimension

2^{n}

over a field

F

which is not similar to any Pfister form. This can be used to show that the implication

(3) \Rightarrow (2)

is not true in characteristic two.…”

Section: Introductionmentioning

confidence: 99%

“…However, by [, (5.5)] an anisotropic symmetric bilinear form of dimension

2^{n}

over

F

, which is either anisotropic or metabolic over all extensions of

F

, is similar to a Pfister form. Considering this result a conjecture may be formulated as follows (see ): Conjecture Let

(A, σ)

be a central simple algebra of degree

2^{n}

with orthogonal involution over a field

F

of characteristic two. If

σ

is anisotropic, then the following statements are equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…The implication

(1) \Rightarrow (2)

follows from [, (6.2)] (even without the anisotropy condition on

σ

). The implication

(2) \Rightarrow (3)

was also proved in [, (8.3)]. The aim of this work is to prove the implication

(3) \Rightarrow (1)

.…”

Section: Introductionmentioning

confidence: 99%

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Pfister involutions in characteristic two

Nokhodkar

2017

Bull. London Math. Soc.

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“…The converse, that any orthogonal involution on an algebra of two-power degree that is anisotropic or hyperbolic after extending scalars to any field extension is necessarily totally decomposable, is clear for split algebras, but otherwise remains largely open. In [7], this question is considered for orthogonal involutions in characteristic two, whose behaviour is somewhat unusual.…”

Section: Introductionmentioning

confidence: 99%

Totally decomposable symplectic and unitary involutions

Dolphin

2016

manuscripta math.

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Abstract. 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 hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.

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Descent properties of biquaternion algebras with involution

Nokhodkar

2017

Math. Nachr.

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Orthogonal Pfister involutions in characteristic two

Cited by 23 publications

References 10 publications

Pfister involutions in characteristic two

Pfister involutions in characteristic two

Totally decomposable symplectic and unitary involutions

Descent properties of biquaternion algebras with involution

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