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) = ∞. 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]

## Contents

* 1 Powers of 2
* 2 Positive characteristic
* 3 Properties
* 4 Examples
* 5 Notes
* 6 References
* 7 Further reading

## Powers of 2

If s(F)≠∞ then s(F)=2k for some natural number k.[1][2]

Proof: Let k∈ℕ be chosen such that 2k≤s(F)<2k+1. Let n=2k. Then there are s=s(F) elements e1,…,es∈F∖{0} such that

0=1+e12+⋯+en−12⏟=:a+en2+⋯+es2⏟=:b.

Both a and b are sums of n squares, and a≠0, since otherwise s(F)<2k, 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=c12+⋯+cn2 for some ci∈F. But since a+b=0, we also have −a2=ab, and hence

−1=aba2=(c1a)2+⋯+(cna)2,

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

## Positive characteristic

Any field F with positive characteristic has s(F)≤2.[3]

Proof: Let p=char⁡(F). It suffices to prove the claim for 𝔽p.

If p=2 then −1=1=12, so s(F)=1.

If p>2 consider the set S={x2:x∈𝔽p} of squares. S∖{0} is a subgroup of index 2 in the cyclic group 𝔽p× with p−1 elements. Thus S contains exactly p+12 elements, and so does −1−S. Since 𝔽p only has p elements in total, S and −1−S cannot be disjoint, that is, there are x,y∈𝔽p with S∋x2=−1−y2∈−1−S and thus −1=x2+y2.

## Properties

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

* The Stufe of a quadratically closed field is 1.[8]
* The Stufe of an algebraic number field is ∞, 1, 2 or 4 (Siegel's theorem).[9] Examples are Q, Q(√−1), Q(√−2) and Q(√−7).[7]
* The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.[3][8][10]
* 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.[9]

## Notes

1. ↑ 1.0 1.1 Rajwade (1993) p.13
2. ↑ Lam (2005) p.379
3. ↑ 3.0 3.1 Rajwade (1993) p.33
4. ↑ Rajwade (1993) p.44
5. ↑ Rajwade (1993) p.228
6. ↑ Lam (2005) p.395
7. ↑ 7.0 7.1 Milnor & Husemoller (1973) p.75
8. ↑ 8.0 8.1 8.2 Lam (2005) p.380
9. ↑ 9.0 9.1 Lam (2005) p.381
10. ↑ Singh, Sahib (1974). "Stufe of a finite field". Fibonacci Quarterly 12: 81–82. ISSN 0015-0517.

## References

* Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2.
* Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X.
* Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. 171. Cambridge University Press. ISBN 0-521-42668-5.

## Further reading

* 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.

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