Stufe (algebra)

In field theory, a branch of mathematics, the Stufe (German pronunciation: [ˈʃtuːfə]; 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) = ${\displaystyle \infty }$. 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 ${\displaystyle s(F)\neq \infty }$ then ${\displaystyle s(F)=2^{k}}$ for some natural number ${\displaystyle k}$.¹²

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

${\displaystyle 0=\underbrace {1+e_{1}^{2}+\cdots +e_{n-1}^{2}} _{=:\,a}+\underbrace {e_{n}^{2}+\cdots +e_{s}^{2}} _{=:\,b}\;.}$

Both ${\displaystyle a}$ and ${\displaystyle b}$ are sums of ${\displaystyle n}$ squares, and ${\displaystyle a\neq 0}$, 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 ${\displaystyle s(F)\leq 2}$.³

Proof: Let ${\displaystyle p=\operatorname {char} (F)}$. 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 ${\displaystyle {\tfrac {p+1}{2}}}$ 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) + 1.⁵⁶ The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).⁷⁸

• The Stufe of a quadratically closed field is 1.⁸
• The Stufe of an algebraic number field is ${\displaystyle \infty }$, 1, 2 or 4 (Siegel's theorem).⁹ Examples are ${\displaystyle \mathbb {Q} }$, ${\displaystyle \mathbb {Q} ({\sqrt {-1}})}$, ${\displaystyle \mathbb {Q} ({\sqrt {-2}})}$ and ${\displaystyle \mathbb {Q} ({\sqrt {-7}})}$.⁷
• The Stufe of a finite field ${\displaystyle \mathbb {F} _{q}}$ is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.³⁸¹⁰
• 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 ${\displaystyle \mathbb {Q} _{2}}$ is 4.⁹