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In [[field theory (mathematics)|field theory]], the '''Stufe''' (/[[Help:IPA for German|ʃtuːfə]]/; German: level) <math>s(F)</math> of a [[field (mathematics)|field]] <math>F</math> is the least number of squares that sum to <math>-1</math>. If <math>-1</math> cannot be written as a sum of squares, <math>s(F)=\infty</math>. In this case, <math>F</math> is a [[formally real field]]. [[Albrecht Pfister (mathematician)|Albrecht Pfister]] proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.<ref name=R13/>
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== Powers of 2 ==
 
If <math>s(F)\ne\infty</math> then <math>s(F)=2^k</math> for some <math>k\in\Bbb N</math>.<ref name=R13>Rajwade (1993) p.13</ref><ref name=Lam379>Lam (2005) p.379</ref>
 
''Proof:'' Let <math>k \in \Bbb N</math> be chosen such that <math>2^k \leq s(F) < 2^{k+1}</math>. Let <math>n = 2^k</math>. Then there are <math>s = s(F)</math> elements <math>e_1, \ldots, e_s \in F\setminus\{0\}</math> such that
 
:<math>0 = \underbrace{1 + e_1^2 + \cdots + e_{n-1}^2 }_{=: a} + \underbrace{e_n^2 + \cdots + e_s^2}_{=: b}\;. </math>
 
Both <math>a</math> and <math>b</math> are sums of <math>n</math> squares, and <math>a \ne 0</math>, since otherwise <math>s(F)\le 2^k</math>, contrary to the assumption on <math>k</math>.
 
According to the theory of [[Pfister form]]s, the product <math>ab</math> is itself a sum of <math>n</math> squares, that is, <math>ab = c_1^2 + \cdots + c_n^2</math> for some <math>c_i \in F</math>. But since <math>a+b=0</math>, we also have <math>-a^2 = ab</math>, and hence
 
:<math>-1 = \frac{ab}{a^2} = \left(\frac{c_1}{a} \right)^2 + \cdots +\left(\frac{c_n}{a} \right)^2\;,</math>
 
and thus <math>s(F) = n = 2^k</math>.
 
== Positive characteristic ==
The Stufe <math>s(F) \le 2</math> for all fields <math>F</math> with positive [[characteristic (algebra)|characteristic]].<ref name=R33>Rajwade (1993) p.33</ref>
 
''Proof:'' Let <math>p = \operatorname{char}(F)</math>. It suffices to prove the claim for <math>\Bbb F_p</math> .
 
If <math>p = 2</math> then <math>-1 = 1 = 1^2</math>, so <math>s(F)=1</math>.
 
If <math>p>2</math> consider the set <math>S=\{x^2\mid x\in\Bbb F_p\}</math> of squares. <math>S\setminus\{0\}</math> is a subgroup of index <math>2</math> in the cyclic group <math>\Bbb F_p^\times</math> with <math>p-1</math> elements. Thus <math>S</math> contains exactly <math>\tfrac{p+1}2</math> elements, and so does <math>-1-S</math>.
Since <math>\Bbb F_p</math> only has <math>p</math> elements in total, <math>S</math> and <math>-1-S</math> cannot be disjoint, that is, there are <math>x,y\in\Bbb F_p</math> with <math>S\ni x^2=-1-y^2\in-1-S</math> and thus <math>-1=x^2+y^2</math>.
 
==Properties==
The Stufe ''s''(''F'') is related to the [[Pythagoras number]] ''p''(''F'') by ''p''(''F'') ≤  ''s''(''F'')+1.<ref name=R44>Rajwade (1993) p.44</ref> If ''F'' is not formally real then ''s''(''F'') ≤ ''p''(''F'') ≤ ''s''(''F'')+1.<ref name=R228>Rajwade (1993) p.228</ref><ref name=Lam395>Lam (2005) p.395</ref>  The additive order of the form (1), and hence the [[Exponent of a group|exponent]] of the [[Witt group]] of ''F'' is equal to 2''s''(''F'').<ref name=MH75>Milnor & Husemoller (1973) p.75</ref><ref name=Lam380>Lam (2005) p.380</ref>
 
==Examples==
* The Stufe of a [[quadratically closed field]] is 1.<ref name=Lam380/>
* The Stufe of an [[algebraic number field]] is ∞, 1, 2 or 4 ("Siegel's theorem).<ref name=Lam381/>  Examples are '''Q''', '''Q'''(√-1), '''Q'''(√-2) and '''Q'''(√-7).<ref name=MH75/>
* The Stufe of a [[finite field]] GF(''q'') is 1 if ''q'' ≡ 1 mod 4 and 2 if ''q'' ≡ 3 mod 4.<ref name=R33/><ref name=Lam380/><ref>{{cite journal | last=Singh | first=Sahib | title=Stufe of a finite field | journal=[[Fibonacci Quarterly]] | volume=12 | pages=81-82 | year=1974 | issn=0015-0517 | zbl=0278.12008 }}</ref>
* 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'''<sub>2</sub> is 4.<ref name=Lam381>Lam (2005) p.381</ref>
 
==Notes==
{{reflist}}
 
==References==
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 }}
* {{cite book | first1=J. | last1=Milnor | author1-link=John Milnor| first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 }}
* {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
 
==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 }}
 
[[Category:Field theory]]

Latest revision as of 16:58, 10 February 2014

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