Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T04:43:11.821Z Has data issue: false hasContentIssue false

COMMON SLOTS OF BILINEAR AND QUADRATIC PFISTER FORMS

Published online by Cambridge University Press:  03 May 2018

ADAM CHAPMAN*
Affiliation:
Department of Computer Science, Tel-Hai Academic College, Upper Galilee, 12208, Israel email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that over any field $F$ of characteristic 2 and 2-rank $n$, there exist $2^{n}$ bilinear $n$-fold Pfister forms that have no slot in common. This answers a question of Becher [‘Triple linkage’, Ann.$K$-Theory, to appear] in the negative. We provide an analogous result also for quadratic Pfister forms.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Arason, J. K. and Baeza, R., ‘Relations in I n and I n W q in characteristic 2’, J. Algebra 314(2) (2007), 895911.Google Scholar
Aravire, R. and Baeza, R., ‘Milnor’s k-theory and quadratic forms over fields of characteristic two’, Comm. Algebra 20(4) (1992), 10871107.Google Scholar
Baeza, R., Quadratic Forms over Semilocal Rings, Lecture Notes in Mathematics, 655 (Springer, New York, 1978).Google Scholar
Baeza, R., ‘Comparing u-invariants of fields of characteristic 2’, Bol. Soc. Brasil. Mat. 13(1) (1982), 105114.Google Scholar
Becher, K. J., ‘Triple linkage’, Ann. $K$ -Theory, to appear.Google Scholar
Chapman, A., ‘Common subfields of p-algebras of prime degree’, Bull. Belg. Math. Soc. Simon Stevin 22(4) (2015), 683686.Google Scholar
Chapman, A. and Dolphin, A., ‘Differential forms, linked fields, and the u-invariant’, Arch. Math. (Basel) 109(2) (2017), 133142.Google Scholar
Chapman, A., Dolphin, A. and Laghribi, A., ‘Total linkage of quaternion algebras and Pfister forms in characteristic two’, J. Pure Appl. Algebra 220(11) (2016), 36763691.Google Scholar
Chapman, A., Dolphin, A. and Leep, D. B., ‘Triple linkage of quadratic Pfister forms’, Manuscripta Math., to appear.Google Scholar
Chapman, A., Gilat, S. and Vishne, U., ‘Linkage of quadratic Pfister forms’, Comm. Algebra 45(12) (2017), 52125226.CrossRefGoogle Scholar
Chapman, A. and McKinnie, K., ‘Kato–Milne cohomology and polynomial forms’, J. Pure Appl. Algebra, to appear.Google Scholar
Draxl, P., ‘Über gemeinsame separabel-quadratische Zerfällungskörper von Quaternionenalgebren’, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 16 (1975), 251259.Google Scholar
Elman, R., Karpenko, N. and Merkurjev, A., The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications, 56 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Elman, R. and Lam, T. Y., ‘Quadratic forms and the u-invariant. II’, Invent. Math. 21 (1973), 125137.Google Scholar
Faivre, F., Liaison des formes de Pfister et corps de fonctions de quadriques en caractéristique 2, PhD Thesis, Université de Franche-Comté, 2006.Google Scholar
Fried, M. D. and Jarden, M., Field Arithmetic, 3rd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, 11 (Springer, Berlin, 2008), revised by M. Jarden.Google Scholar
Kato, K., ‘Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two’, Invent. Math. 66(3) (1982), 493510.Google Scholar
Lam, T. Y., ‘On the linkage of quaternion algebras’, Bull. Belg. Math. Soc. Simon Stevin 9(3) (2002), 415418.Google Scholar
Tignol, J.-P. and Wadsworth, A. R., Value Functions on Simple Algebras, and Associated Graded Rings, Springer Monographs in Mathematics (Springer, Cham, 2015).Google Scholar