Pythagoras number

Last updated August 25, 2024

In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p(K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares.

Examples

Every non-negative real number is a square, so p(R) = 1.
For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares,^[1] so p = 2.
By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so p(Q) = 4.

Properties

Every positive integer occurs as the Pythagoras number of some formally real field.^[2]
The Pythagoras number is related to the Stufe by p(F) ≤ s(F) + 1.^[3] If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1,^[4] and both cases are possible: for F = C we have s = p = 1, whereas for F = F₅ we have s = 1, p = 2.^[5]
As a consequence, the Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair (s,p) of the form (2^k,2^k) or (2^k,2^k + 1), there exists a field F such that (s(F),p(F)) = (s,p).^[6] For example, quadratically closed fields (e.g., C) and fields of characteristic 2 (e.g., F₂) give (s(F),p(F)) = (1,1); for primes p ≡ 1 (mod 4), F_p and the p-adic field Q_p give (1,2); for primes p ≡ 3 (mod 4), F_p gives (2,2), and Q_p gives (2,3); Q₂ gives (4,4), and the function field Q₂(X) gives (4,5).
The Pythagoras number is related to the height of a field F: if F is formally real then h(F) is the smallest power of 2 which is not less than p(F); if F is not formally real then h(F) = 2s(F).^[7]

Notes

↑ Lam (2005) p. 36
↑ Lam (2005) p. 398
↑ Rajwade (1993) p. 44
↑ Rajwade (1993) p. 228
↑ Rajwade (1993) p. 261
↑ Lam (2005) p. 396
↑ Lam (2005) p. 395

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References

Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.

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[Lam36-1] Lam (2005) p. 36

[Lam398-2] Lam (2005) p. 398

[R44-3] Rajwade (1993) p. 44

[R228-4] Rajwade (1993) p. 228

[R261-5] Rajwade (1993) p. 261

[Lam396-6] Lam (2005) p. 396

[Lam395-7] Lam (2005) p. 395

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