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) = ${\displaystyle \mathrm{\infty }}$. 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.^[1]

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

Proof: Let ${\displaystyle k\in \mathbb{\mathbb{N}}}$ be chosen such that ${\displaystyle 2^k\le s(F)<2^{k+1}}$. Let ${\displaystyle n=2^k}$. Then there are ${\displaystyle s=s(F)}$ elements ${\displaystyle e_1,\dots ,e_s\in F\setminus \{0\}}$ such that

${\displaystyle 0=\underset{=:a}{\underbrace{1+e_1^2+\cdots +e_{n-1}^2}}+\underset{=:b}{\underbrace{e_n^2+\cdots +e_s^2}}.}$

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

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

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

and thus ${\displaystyle s(F)=n=2^k}$.

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

Proof: Let ${\displaystyle p=\mathrm{char}(F)}$. It suffices to prove the claim for ${\displaystyle {\mathbb{𝔽}}_p}$.

If ${\displaystyle p=2}$ then ${\displaystyle -1=1=1^2}$, so ${\displaystyle s(F)=1}$.

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

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]

• 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]

• 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. 

• 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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