Stufe (algebra) explained

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) =

inf ty . 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 inf ty then s(F) =2 ^{k} for some natural number k .^{[1] [2]}

Proof: Let

k \in N be chosen such that 2 ^{k} \leq s(F) < 2 ^{k+1} . Let n = 2 ^{k} . Then there are s = s(F) elements e _{1,} \ldots, e _{s} \in F\setminus\{ 0 \} such that0 = \underbrace{ 1 + e ^{2} _{1} + … + e ^{2} _{n -1} } _{=:a} + \underbrace{e ^{2} _{n} + … + e ^{2 }} _{= :b} .

Both

a and b are sums of n squares, and a \ne 0 , since otherwise s(F)< 2 ^{k} , 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 ^{2} _{1} + … + c ^{2} _{n} for some c _{i} \in F . But since a + b =0 , we also have - a ^{2} = ab , and hence-1 = (ab)/(a ^{2}) = \left( (c _{1})/(a) \right) ^{2} + … + \left( (c _{n})/(a) \right) ^{2,}

and thus

s(F) = n = 2 ^{k} .

Positive characteristic

Any field

F with positive characteristic has s(F) \le 2 .^[3]

Proof: Let

p = \operatorname{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 \in F _{p\}} of squares. S\setminus\{ 0 \} is a subgroup of index 2 in the cyclic group F ^{x} _{p} with p -1 elements. Thus S contains exactly \tfrac{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\in 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.^[4] If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1.^{[5] [6]} The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).^{[7] [8]}

Examples

• The Stufe of a quadratically closed field is 1.^[8]
• 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).^[7]
• The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.^{[3] [8] [9]}
• 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.^[10]

References

• Book: Lam, Tsit-Yuen . Tsit Yuen Lam

. Introduction to Quadratic Forms over Fields . 67 . . Tsit Yuen Lam . American Mathematical Society . 2005 . 0-8218-1095-2 . 1068.11023 .

• Book: J. . Milnor . John Milnor. D. . Husemoller . Symmetric Bilinear Forms . . 73 . . 1973 . 3-540-06009-X . 0292.10016 .
• Book: Rajwade, A. R. . Squares . 171 . London Mathematical Society Lecture Note Series . . 1993 . 0-521-42668-5 . 0785.11022 .

Further reading

• Book: Knebusch . Manfred . Scharlau . Winfried . Algebraic theory of quadratic forms. Generic methods and Pfister forms . . DMV Seminar . 1 . Boston - Basel - Stuttgart . Birkhäuser Verlag . 1980 . 3-7643-1206-8 . 0439.10011 .

Notes and References

1. Rajwade (1993) p.13
2. Lam (2005) p.379
3. Rajwade (1993) p.33
4. Rajwade (1993) p.44
5. Rajwade (1993) p.228
6. Lam (2005) p.395
7. Milnor & Husemoller (1973) p.75
8. Lam (2005) p.380
9. Singh . Sahib . Stufe of a finite field . . 12 . 81–82 . 1974 . 0015-0517 . 0278.12008 .
10. Lam (2005) p.381