Stufe (algebra)

In field theory, 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 $s(F)\ne\infty$ then $s(F)=2^k$ for some $k\in\Bbb N$ .^[1]^[2]

Proof: Let $k \in \Bbb N$ 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 $a \ne 0$ , since otherwise $s(F)\le 2^k$ , 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

The Stufe $s(F) \le 2$ for all fields $F$ with positive characteristic.^[3]

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

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

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

↑ 1.0 1.1 Rajwade (1993) p.13
↑ Lam (2005) p.379
↑ 3.0 3.1 Rajwade (1993) p.33
↑ Rajwade (1993) p.44
↑ Rajwade (1993) p.228
↑ Lam (2005) p.395
↑ 7.0 7.1 Milnor & Husemoller (1973) p.75
↑ 8.0 8.1 8.2 Lam (2005) p.380
↑ 9.0 9.1 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 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 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.