Stufe (algebra)

In field theory, a branch of mathematics, the Stufe (; 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