Level (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 $k$.Rajwade (1993) p.13Lam (2005) p.379

''Proof:'' Let $k \in \mathbb N$ be chosen such that $2^k \leq s(F) < 2^$. Let $n = 2^k$. Then there are $s = s(F)$ elements $e_1, \ldots, e_s \in F\setminus\$ such that

:$0 = \underbrace_ + \underbrace_\;.$

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

:$-1 = \frac = \left(\frac \right)^2 + \cdots + \left(\frac \right)^2,$

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

Positive characteristic

Any field $F$ with positive characteristic has $s(F) \le 2$.Rajwade (1993) p.33

''Proof:'' Let $p = \operatorname(F)$. 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=\$ of squares. $S\setminus\$ is a subgroup of index $2$ in the cyclic group $\mathbb F_p^\times$ with $p-1$ elements. Thus $S$ contains exactly $\tfrac2$ 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.Rajwade (1993) p.44 If ''F'' is not formally real then ''s''(''F'') ≤ ''p''(''F'') ≤ ''s''(''F'') + 1.Rajwade (1993) p.228Lam (2005) p.395 The additive order of the form (1), and hence the exponent of the Witt group of ''F'' is equal to 2''s''(''F'').Milnor & Husemoller (1973) p.75Lam (2005) p.380

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.Lam (2005) p.381

Notes

References

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Further reading

* {{cite book , last1=Knebusch , first1=Manfred , last2=Scharlau , first2=Winfried , title=Algebraic theory of quadratic forms. Generic methods and Pfister forms , others=Notes taken by Heisook Lee , series=DMV Seminar , volume=1 , location=Boston - Basel - Stuttgart , publisher=Birkhäuser Verlag , year=1980 , isbn=3-7643-1206-8 , zbl=0439.10011

Field (mathematics)