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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On quadratic forms of height two
and a theorem of Wadsworth


Author: Detlev W. Hoffmann
Journal: Trans. Amer. Math. Soc. 348 (1996), 3267-3281
MSC (1991): Primary 11E04, 11E81, 12F20
MathSciNet review: 1355298
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Abstract: Let $\varphi $ and $\psi $ be anisotropic quadratic forms over a field $F$ of characteristic not $2$. Their function fields $F(% \varphi )$ and $F(\psi )$ are said to be equivalent (over $F$) if $% \varphi \otimes F(\psi )$ and $\psi \otimes F(% \varphi )$ are isotropic. We consider the case where $\dim % \varphi =2^n$ and $% \varphi $ is divisible by an $(n-2)$-fold Pfister form. We determine those forms $\psi $ for which $% \varphi $ becomes isotropic over $F(\psi )$ if $n\leq 3$, and provide partial results for $n\geq 4$. These results imply that if $F(% \varphi )$ and $F(\psi )$ are equivalent and $\dim % \varphi =\dim \psi$, then $% \varphi $ is similar to $\psi $ over $F$. This together with already known results yields that if $% \varphi $ is of height $2$ and degree $1$ or $2$, and if $\dim % \varphi =\dim \psi$, then $F(% \varphi )$ and $F(\psi )$ are equivalent iff $F(% \varphi )$ and $F(\psi )$ are isomorphic over $F$.


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Additional Information

Detlev W. Hoffmann
Affiliation: Aindorferstr. 84, D-80689 Munich, Germany
Address at time of publication: Laboratoire de Mathématiques, Faculté des Sciences, Université de Franche-Comté, 25030 Besançon Cedex, France
Email: detlev@math.univ-fcomte.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01637-6
PII: S 0002-9947(96)01637-6
Keywords: Quadratic forms of height 2, function fields of quadratic forms, equivalence of function fields, isomorphism of function fields
Received by editor(s): December 2, 1994
Received by editor(s) in revised form: October 16, 1995
Additional Notes: This research has been carried out during the author’s stay at the Department of Mathematics at the University of Kentucky, Lexington, Kentucky, during the academic year 1994/95.
Article copyright: © Copyright 1996 American Mathematical Society