Stufe (algebra)

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In field theory, 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[edit]

If then for some .[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[edit]

The Stufe for all fields with positive characteristic.[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[edit]

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[edit]

Notes[edit]

  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[edit]

Further reading[edit]

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