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) = $\infty$. 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.

Powers of 2

If $s(F)\neq \infty$ then $s(F)=2^{k}$ for some natural number $k$.

Proof: Let $k\in \mathbb {N}$ 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 that

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

Both $a$ and $b$ are sums of $n$ squares, and $a\neq 0$, 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, $ab=c_{1}^{2}+\cdots +c_{n}^{2}$ for some $c_{i}\in F$. But since $a+b=0$, we also have $-a^{2}=ab$, and hence

$-1={\frac {ab}{a^{2}}}=\left({\frac {c_{1}}{a}}\right)^{2}+\cdots +\left({\frac {c_{n}}{a}}\right)^{2},$

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

Positive characteristic

Any field $F$ with positive characteristic has $s(F)\leq 2$.

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

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

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

Examples

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

References

• Knebusch, Manfred; Scharlau, Winfried (1980). Algebraic Theory of Quadratic Forms. Generic Methods and Pfister Forms. DMV Seminar. Vol. 1. Notes taken by Heisook Lee. Basel: Birkhäuser. ISBN 3-7643-1206-8. Zbl 0439.10011.

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.

• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. A Series of Modern Surveys in Mathematics. Vol. 73. Berlin, Heidelberg: Springer. ISBN 3-540-06009-X. Zbl 0292.10016.

• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.

• Singh, Sahib (1974). "Stufe of a finite field". Fibonacci Quarterly. 12 (1): 81–82. doi:10.1080/00150517.1974.12430777. Zbl 0278.12008.