Stufe (algebra)

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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) = . 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.[1]

Powers of 2

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If   then   for some natural number  .[1][2]

Proof: Let   be chosen such that  . Let  . Then there are   elements   such that

 

Both   and   are sums of   squares, and  , since otherwise  , contrary to the assumption on  .

According to the theory of Pfister forms, the product   is itself a sum of   squares, that is,   for some  . But since  , we also have  , and hence

 

and thus  .

Positive characteristic

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Any field   with positive characteristic has  .[3]

Proof: Let  . It suffices to prove the claim for  .

If   then  , so  .

If   consider the set   of squares.   is a subgroup of index   in the cyclic group   with   elements. Thus   contains exactly   elements, and so does  . Since   only has   elements in total,   and   cannot be disjoint, that is, there are   with   and thus  .

Properties

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The Stufe s(F) is related to the Pythagoras number p(F) by p(F) ≤ s(F) + 1.[4] If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1.[5][6] The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).[7][8]

Examples

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Notes

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  1. ^ a b Rajwade (1993) p.13
  2. ^ Lam (2005) p.379
  3. ^ a b Rajwade (1993) p.33
  4. ^ Rajwade (1993) p.44
  5. ^ Rajwade (1993) p.228
  6. ^ Lam (2005) p.395
  7. ^ a b Milnor & Husemoller (1973) p.75
  8. ^ a b c Lam (2005) p.380
  9. ^ a b Lam (2005) p.381
  10. ^ Singh, Sahib (1974). "Stufe of a finite field". Fibonacci Quarterly. 12: 81–82. ISSN 0015-0517. Zbl 0278.12008.

References

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Further reading

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  • Knebusch, Manfred; Scharlau, Winfried (1980). Algebraic theory of quadratic forms. Generic methods and Pfister forms. DMV Seminar. Vol. 1. Notes taken by Heisook Lee. Boston - Basel - Stuttgart: Birkhäuser Verlag. ISBN 3-7643-1206-8. Zbl 0439.10011.