In field theory, 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.
If s ( F ) ≠ ∞ then s ( F ) = 2 k for some k ∈ N .
Proof: Let k ∈ N be chosen such that 2 k ≤ s ( F ) < 2 k + 1 . Let n = 2 k . Then there are s = s ( F ) elements e 1 , … , e s ∈ F ∖ { 0 } such that
0 = 1 + e 1 2 + ⋯ + e n − 1 2 ⏟ =: a + e n 2 + ⋯ + e s 2 ⏟ =: b . Both a and b are sums of n squares, and a ≠ 0 , since otherwise s ( F ) < 2 k , contrary to the assumption on k .
According to the theory of Pfister forms, the product a b is itself a sum of n squares, that is, a b = c 1 2 + ⋯ + c n 2 for some c i ∈ F . But since a + b = 0 , we also have − a 2 = a b , and hence
− 1 = a b a 2 = ( c 1 a ) 2 + ⋯ + ( c n a ) 2 , and thus s ( F ) = n = 2 k .
The Stufe s ( F ) ≤ 2 for all fields F with positive characteristic.
Proof: Let p = char ( F ) . It suffices to prove the claim for F p .
If p = 2 then − 1 = 1 = 1 2 , so s ( F ) = 1 .
If p > 2 consider the set S = { x 2 ∣ x ∈ F p } of squares. S ∖ { 0 } is a subgroup of index 2 in the cyclic group F p × with p − 1 elements. Thus S contains exactly p + 1 2 elements, and so does − 1 − S . Since F p only has p elements in total, S and − 1 − S cannot be disjoint, that is, there are x , y ∈ F p with S ∋ x 2 = − 1 − y 2 ∈ − 1 − S and thus − 1 = x 2 + y 2 .
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).
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.