Overview
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) = \infty . 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 s(F)\ne\infty then s(F)=2^k for some natural number k . Proof: Let k \in \mathbb N be chosen such that 2^k \leq s(F) . Let n = 2^k . Then there are s = s(F) elements e_1, \ldots, e_s \in F\setminus\ 0\ such that : 0 = \underbrace 1 + e_1^2 + \cdots + e_ n-1 ^2 _ =:\,a + \underbrace e_n^2 + \cdots + e_s^2 _ =:\,b \;. Both a and b are sums of n squares, and a \ne 0 , since otherwise s(F) , contrary to the assumption on k . According to the theory of Pfister forms, the product ab is itself a sum of n squares, that is, ab = c_1^2 + \cdots + c_n^2 for some c_i \in F . But since a+b=0 , we also have -a^2 = ab , and hence : -1 = \frac ab a^2 = \left(\frac c_1 a \right)^2 + \cdots + \left(\frac c_n a \right)^2, and thus s(F) = n = 2^k .
Positive characteristic
Any field F with positive characteristic has s(F) \leq 2 . Proof: Let p = \operatorname char (F) . It suffices to prove the claim for \mathbb F_p . If p = 2 then -1 = 1 = 1^2 , so s(F)=1 . If p>2 consider the set S=\ x^2 : x \in \mathbb F_p\ of squares. S\setminus\ 0\ is a subgroup of index 2 in the cyclic group \mathbb F_p^\times with p-1 elements. Thus S contains exactly \tfrac p+1 2 elements, and so does -1-S . Since \mathbb F_p only has p elements in total, S and -1-S cannot be disjoint, that is, there are x,y\in\mathbb F_p with S\ni x^2=-1-y^2\in-1-S and thus -1=x^2+y^2 .
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 \infty , 1, 2 or 4 (Siegel's theorem). Examples are \Q , \Q(\sqrt -1 ) , \Q(\sqrt -2 ) and \Q(\sqrt -7 ) . * The Stufe of a finite field \mathbb F _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 \Q_2 is 4.