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The $u^n$-invariant and the symbol length of $H_2^n(F)$


Authors: Adam Chapman and Kelly McKinnie
Journal: Proc. Amer. Math. Soc. 147 (2019), 513-521
MSC (2010): Primary 11E81; Secondary 11E04, 12G05
DOI: https://doi.org/10.1090/proc/14308
Published electronically: November 8, 2018
MathSciNet review: 3894891
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Abstract: Given a field $F$ of $\operatorname {char}(F)=2$, we define $u^n(F)$ to be the maximal dimension of an anisotropic form in $I_q^n F$. For $n=1$ it recaptures the definition of $u(F)$. We study the relations between this value and the symbol length of $H_2^n(F)$, denoted by $sl_2^n(F)$. We show for any $n \geqslant 2$ that if $2^n \leqslant u^n(F) \leqslant u^2(F) < \infty$, then $sl_2^n(F) \leqslant \prod _{i=2}^n (\frac {u^i(F)}{2}+1-2^{i-1})$. As a result, if $u(F)$ is finite, then $sl_2^n(F)$ is finite for any $n$, a fact which was previously proven when $\operatorname {char}(F) \neq 2$ by Saltman and Krashen. We also show that if $sl_2^n(F)=1$, then $u^n(F)$ is either $2^n$ or $2^{n+1}$.


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Additional Information

Adam Chapman
Affiliation: Department of Computer Science, Tel-Hai Academic College, Upper Galilee, 12208 Israel
MR Author ID: 983980
Email: adam1chapman@yahoo.com

Kelly McKinnie
Affiliation: Department of Mathematics, University of Montana, Missoula, Montana 59812
MR Author ID: 825554
Email: kelly.mckinnie@mso.umt.edu

Keywords: Kato-Milne cohomology, quadratic forms, symbol length, $u$-invariant
Received by editor(s): September 15, 2017
Received by editor(s) in revised form: March 27, 2018
Published electronically: November 8, 2018
Additional Notes: The first author was visiting Perimeter Institute for Theoretical Physics during the summer of 2017, during which a significant portion of the work on this project was carried out.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2018 American Mathematical Society