Supreme Pfister Forms

Abstract

For a nonreal field F of characteristic different from 2, we compare several properties which F may have with respect to an anisotropic n-fold Pfister form π over F. In particular, we consider the situation where the subforms of π are all the anisotropic quadratic forms over F; then π is said to be supreme. We further apply our results to the case where π is the form 2 ⁿ  × ⟨1⟩ (n ≥ 1). In this way we obtain new examples of nonreal fields with prescribed level and with additional properties.

Key Words:

• Pfister form
• local field
• Hilbert symbol
• nonreal field
• level
• symbol length

Acknowledgments

The author wishes to express his gratitude to Detlev Hoffmann, Ahmed Laghribi and Jean-Pierre Tignol for various remarks, which had an important influence on the development of this research. He also greatfully thanks Tatiana Beliaeva and Thomas Unger for many valuable comments, which helped improve the manuscript.

This research was started during the author's doctoral studies at Université de Franche-Comté (Besançon, France). It was continued and finished during two postdoctoral visits, at Université Catholique de Louvain (Louvain-la-Neuve, Belgium), and at University College Dublin (Ireland), both financed by the TMR network “Algebraic K-Theory, Linear Algebraic Groups and Related Structures” (ERB FMRX CT-97-0107). The author wishes to thank these institutions for their hospitality and support.

Notes

^#Communicated by A. Prestel.