In field theory, 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.
Powers of 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 .
The Stufe for all fields with positive characteristic.
Proof: Let . It suffices to prove the claim for .
If then , so .
If consider the set of squares. is a subgroup of index in the cyclic group with elements. Thus contains exactly elements, and so does . Since only has elements in total, and cannot be disjoint, that is, there are with and thus .
The Stufe s(F) is related to the Pythagoras number p(F) by p(F) ≤ s(F)+1. If F is not formally real then s(F) ≤ p(F) ≤ s(F)+1. The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).
- The Stufe of a quadratically closed field is 1.
- The Stufe of an algebraic number field is ∞, 1, 2 or 4 ("Siegel's theorem). Examples are Q, Q(√-1), Q(√-2) and Q(√-7).
- The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.
- The Stufe of a local field of odd residue characteristic is equal to that of its residue field. The Stufe of the 2-adic field Q2 is 4.
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- Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
- Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.