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 .
Any field with positive characteristic has . [3]
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. [4] If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1. [5] [6] The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F). [7] [8]
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 .
Any field with positive characteristic has . [3]
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. [4] If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1. [5] [6] The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F). [7] [8]