Stufe (algebra)


In field theory, the Stufe s 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=. 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.
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

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.

Properties

The Stufe s is related to the Pythagoras number p by p ≤ s+1. If F is not formally real then s ≤ p ≤ s+1. The additive order of the form, and hence the exponent of the Witt group of F is equal to 2s.

Examples