In field theory, a branch of mathematics, the Stufe (/ʃtuːfə/; German: level) s(F) of a field F is the least number of squares that sum to −1. If −1 cannot be written as a sum of squares, s(F) = . In this case, F is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.[1]
If then for some natural number .[1][2]
Proof: Let be chosen such that . Let . Then there are elements such that
Both and are sums of squares, and , since otherwise , contrary to the assumption on .
According to the theory of Pfister forms, the product is itself a sum of squares, that is, for some . But since , we also have , and hence
and thus .
Milnor & Husemoller (1973) p.75
- Knebusch, Manfred; Scharlau, Winfried (1980). Algebraic theory of quadratic forms. Generic methods and Pfister forms. DMV Seminar. Vol. 1. Notes taken by Heisook Lee. Boston - Basel - Stuttgart: Birkhäuser Verlag. ISBN 3-7643-1206-8. Zbl 0439.10011.