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.

Powers of 2
If $$s(F)\ne\infty$$ 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 $$a \ne 0$$, 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 $$s(F) \le 2$$.

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 ∞, 1, 2 or 4 (Siegel's theorem). Examples are Q, Q(√−1), Q(√−2) and Q(√−7).
 * The Stufe of a finite field GF(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 Q2 is 4.