Pythagoras number

Last updated November 10, 2020

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]
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).^[6] As a consequence, the Pythagoras number of a non-formally-real field, if finite, is either a power of 2 or 1 less than a power of 2, and all cases occur.^[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. 395
↑ Lam (2005) p. 396

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References

Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 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. 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

[Lam395-6] Lam (2005) p. 395

[Lam396-7] Lam (2005) p. 396