Pythagoras number

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.

A Pythagorean field is one with Pythagoras number 1: that is, every sum of squares is already a square.

Examples

• Every positive real 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

1. Lam (2005) p. 36
2. Lam (2005) p. 398
3. Rajwade (1993) p. 44
4. Rajwade (1993) p. 228
5. Rajwade (1993) p. 261
6. Lam (2005) p. 395
7. Lam (2005) p. 396

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.