Stufe (algebra)

In field theory, a branch of mathematics, the Stufe (Template:IPA; 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.Template:Sfnp

Powers of 2

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

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

0 = \underset{= : a}{\underset{⏟}{1 + e_{1}^{2} + \dots + e_{n - 1}^{2}}} + \underset{= : b}{\underset{⏟}{e_{n}^{2} + \dots + e_{s}^{2}}} .

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 $a b$ is itself a sum of $n$ squares, that is, $a b = c_{1}^{2} + \dots + c_{n}^{2}$ for some $c_{i} \in F$ . But since $a + b = 0$ , we also have $- a^{2} = a b$ , and hence

- 1 = \frac{a b}{a^{2}} = {(\frac{c_{1}}{a})}^{2} + \dots + {(\frac{c_{n}}{a})}^{2},

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

Positive characteristic

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

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

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

If $p > 2$ consider the set $S = {x^{2} : x \in 𝔽_{p}}$ of squares. $S ∖ {0}$ is a subgroup of index $2$ in the cyclic group $𝔽_{p}^{\times}$ with $p - 1$ elements. Thus $S$ contains exactly $\frac{p + 1}{2}$ elements, and so does $- 1 - S$ . Since $𝔽_{p}$ only has $p$ elements in total, $S$ and $- 1 - S$ cannot be disjoint, that is, there are $x, y \in 𝔽_{p}$ with $S ∋ x^{2} = - 1 - y^{2} \in - 1 - S$ 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.Template:Sfnp If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1.Template:Sfnp Template:Sfnp The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).Template:Sfnp Template:Sfnp

Examples

The Stufe of a quadratically closed field is 1.Template:Sfnp
The Stufe of an algebraic number field is $\infty$ , 1, 2 or 4 (Siegel's theorem).Template:Sfnp Examples are $ℚ$ , $ℚ (\sqrt{- 1})$ , $ℚ (\sqrt{- 2})$ and $ℚ (\sqrt{- 7})$ .Template:Sfnp
The Stufe of a finite field $𝔽_{q}$ is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.Template:Sfnp Template:Sfnp Template:Sfnp
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 $ℚ_{2}$ is 4.Template:Sfnp