The u                         n                         -invariant and the symbol length of H                         n                                                                              2                          (F)

Adam Chapman; Kelly McKinnie

doi:10.1090/proc/14308

The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F)

Adam Chapman, Kelly McKinnie

Mathematics

Tel Hai Academic College

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Given a field F of char(F) = 2, we define u ⁿ (F) to be the maximal dimension of an anisotropic form in Iq ⁿ F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of H ⁿ ₂ (F), denoted by sl ⁿ ₂ (F). We show for any n ≥ 2 that if 2 ⁿ ≤ u ⁿ (F) ≤ u ² (F) < ∞, then (Formula Presented). As a result, if u(F) is finite, then sl ⁿ ₂ (F) is finite for any n, a fact which was previously proven when char(F) ≠ 2 by Saltman and Krashen. We also show that if sl ⁿ ₂ (F) = 1, then u ⁿ (F) is either 2 ⁿ or 2 ⁿ⁺¹ .

Original language	English
Pages (from-to)	513-521
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	147
Issue number	2
DOIs	https://doi.org/10.1090/proc/14308
State	Published - Feb 2019

Keywords

Kato-Milne cohomology
Quadratic forms
Symbol length
U-invariant

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10.1090/proc/14308

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Chapman, A., & McKinnie, K. (2019). The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F) Proceedings of the American Mathematical Society, 147(2), 513-521. https://doi.org/10.1090/proc/14308

Chapman, Adam ; McKinnie, Kelly. / The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F) In: Proceedings of the American Mathematical Society. 2019 ; Vol. 147, No. 2. pp. 513-521.

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title = " The u n -invariant and the symbol length of H n 2 (F) ",

abstract = " Given a field F of char(F) = 2, we define u n (F) to be the maximal dimension of an anisotropic form in Iq n F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of H n 2 (F), denoted by sl n 2 (F). We show for any n ≥ 2 that if 2 n ≤ u n (F) ≤ u 2 (F) < ∞, then (Formula Presented). As a result, if u(F) is finite, then sl n 2 (F) is finite for any n, a fact which was previously proven when char(F) ≠ 2 by Saltman and Krashen. We also show that if sl n 2 (F) = 1, then u n (F) is either 2 n or 2 n+1 .",

keywords = "Kato-Milne cohomology, Quadratic forms, Symbol length, U-invariant",

author = "Adam Chapman and Kelly McKinnie",

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Chapman, A & McKinnie, K 2019, ' The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F) ', Proceedings of the American Mathematical Society, vol. 147, no. 2, pp. 513-521. https://doi.org/10.1090/proc/14308

The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F) . / Chapman, Adam; McKinnie, Kelly.
In: Proceedings of the American Mathematical Society, Vol. 147, No. 2, 02.2019, p. 513-521.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - The u n -invariant and the symbol length of H n 2 (F)

AU - Chapman, Adam

AU - McKinnie, Kelly

PY - 2019/2

Y1 - 2019/2

N2 - Given a field F of char(F) = 2, we define u n (F) to be the maximal dimension of an anisotropic form in Iq n F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of H n 2 (F), denoted by sl n 2 (F). We show for any n ≥ 2 that if 2 n ≤ u n (F) ≤ u 2 (F) < ∞, then (Formula Presented). As a result, if u(F) is finite, then sl n 2 (F) is finite for any n, a fact which was previously proven when char(F) ≠ 2 by Saltman and Krashen. We also show that if sl n 2 (F) = 1, then u n (F) is either 2 n or 2 n+1 .

AB - Given a field F of char(F) = 2, we define u n (F) to be the maximal dimension of an anisotropic form in Iq n F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of H n 2 (F), denoted by sl n 2 (F). We show for any n ≥ 2 that if 2 n ≤ u n (F) ≤ u 2 (F) < ∞, then (Formula Presented). As a result, if u(F) is finite, then sl n 2 (F) is finite for any n, a fact which was previously proven when char(F) ≠ 2 by Saltman and Krashen. We also show that if sl n 2 (F) = 1, then u n (F) is either 2 n or 2 n+1 .

KW - Kato-Milne cohomology

KW - Quadratic forms

KW - Symbol length

KW - U-invariant

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DO - 10.1090/proc/14308

M3 - Article

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SN - 0002-9939

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Chapman A, McKinnie K. The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F) Proceedings of the American Mathematical Society. 2019 Feb;147(2):513-521. doi: 10.1090/proc/14308

The u ⁿ -invariant and the symbol length of H ⁿ ₂ (F)

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