Stufe (algebra)

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) = [math]\displaystyle{ \infty }[/math]. 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.^[1]

Powers of 2

If [math]\displaystyle{ s(F)\ne\infty }[/math] then [math]\displaystyle{ s(F)=2^k }[/math] for some natural number [math]\displaystyle{ k }[/math].^[1][2]

Proof: Let [math]\displaystyle{ k \in \mathbb N }[/math] be chosen such that [math]\displaystyle{ 2^k \leq s(F) \lt 2^{k+1} }[/math]. Let [math]\displaystyle{ n = 2^k }[/math]. Then there are [math]\displaystyle{ s = s(F) }[/math] elements [math]\displaystyle{ e_1, \ldots, e_s \in F\setminus\{0\} }[/math] such that

[math]\displaystyle{ 0 = \underbrace{1 + e_1^2 + \cdots + e_{n-1}^2 }_{=:\,a} + \underbrace{e_n^2 + \cdots + e_s^2}_{=:\,b}\;. }[/math]

Both [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are sums of [math]\displaystyle{ n }[/math] squares, and [math]\displaystyle{ a \ne 0 }[/math], since otherwise [math]\displaystyle{ s(F)\lt 2^k }[/math], contrary to the assumption on [math]\displaystyle{ k }[/math].

According to the theory of Pfister forms, the product [math]\displaystyle{ ab }[/math] is itself a sum of [math]\displaystyle{ n }[/math] squares, that is, [math]\displaystyle{ ab = c_1^2 + \cdots + c_n^2 }[/math] for some [math]\displaystyle{ c_i \in F }[/math]. But since [math]\displaystyle{ a+b=0 }[/math], we also have [math]\displaystyle{ -a^2 = ab }[/math], and hence

[math]\displaystyle{ -1 = \frac{ab}{a^2} = \left(\frac{c_1}{a} \right)^2 + \cdots + \left(\frac{c_n}{a} \right)^2, }[/math]

and thus [math]\displaystyle{ s(F) = n = 2^k }[/math].

Positive characteristic

Any field [math]\displaystyle{ F }[/math] with positive characteristic has [math]\displaystyle{ s(F) \le 2 }[/math].^[3]

Proof: Let [math]\displaystyle{ p = \operatorname{char}(F) }[/math]. It suffices to prove the claim for [math]\displaystyle{ \mathbb F_p }[/math].

If [math]\displaystyle{ p = 2 }[/math] then [math]\displaystyle{ -1 = 1 = 1^2 }[/math], so [math]\displaystyle{ s(F)=1 }[/math].

If [math]\displaystyle{ p\gt 2 }[/math] consider the set [math]\displaystyle{ S=\{x^2 : x \in \mathbb F_p\} }[/math] of squares. [math]\displaystyle{ S\setminus\{0\} }[/math] is a subgroup of index [math]\displaystyle{ 2 }[/math] in the cyclic group [math]\displaystyle{ \mathbb F_p^\times }[/math] with [math]\displaystyle{ p-1 }[/math] elements. Thus [math]\displaystyle{ S }[/math] contains exactly [math]\displaystyle{ \tfrac{p+1}2 }[/math] elements, and so does [math]\displaystyle{ -1-S }[/math]. Since [math]\displaystyle{ \mathbb F_p }[/math] only has [math]\displaystyle{ p }[/math] elements in total, [math]\displaystyle{ S }[/math] and [math]\displaystyle{ -1-S }[/math] cannot be disjoint, that is, there are [math]\displaystyle{ x,y\in\mathbb F_p }[/math] with [math]\displaystyle{ S\ni x^2=-1-y^2\in-1-S }[/math] and thus [math]\displaystyle{ -1=x^2+y^2 }[/math].

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).^[9] 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][10]
• 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.^[9]

Notes

References

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2. 
• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X. 
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. 171. Cambridge University Press. ISBN 0-521-42668-5. 

Further reading

• Knebusch, Manfred; Scharlau, Winfried (1980). Algebraic theory of quadratic forms. Generic methods and Pfister forms. DMV Seminar. 1. Notes taken by Heisook Lee. Boston - Basel - Stuttgart: Birkhäuser Verlag. ISBN 3-7643-1206-8. 

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