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

Proof: Let $k\in \mathbb{N}$ be chosen such that $2^k\le s(F)<2^{k+1}$. Let $n=2^k$. Then there are $s=s(F)$ elements $e_1,\dots ,e_s\in F\setminus \{0\}$ such that

$0=\underset{=:a}{\underbrace{1+e_1^2+\cdots +e_{n-1}^2}}+\underset{=:b}{\underbrace{e_n^2+\cdots +e_s^2}}.$

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 $p=\mathrm{char}(F)$. 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. $S\setminus \{0\}$ 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 $S\ni x^2=-1-y^2\in -1-S$ 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.