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) = $\infty$ . 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]

Powers of 2

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

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 $s(F)\leq 2$ .^[3]

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.^[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

^ ^a ^b Rajwade (1993) p.13
^ Lam (2005) p.379
^ ^a ^b Rajwade (1993) p.33
^ Rajwade (1993) p.44
^ Rajwade (1993) p.228
^ Lam (2005) p.395
^ ^a ^b Milnor & Husemoller (1973) p.75
^ ^a ^b ^c Lam (2005) p.380
^ ^a ^b Lam (2005) p.381
^ Singh, Sahib (1974). "Stufe of a finite field". Fibonacci Quarterly. 12: 81–82. ISSN 0015-0517. Zbl 0278.12008.

References

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. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. 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.