Stufe (algebra)
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. Powers of 2

If  then  for some natural number . 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 . Positive characteristic
Any field  with positive characteristic has . 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 . Properties
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). Examples
 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. Notes

References

Further reading
 

Field (mathematics)