Pythagoras number explained

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

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]

References

• Book: Lam, Tsit-Yuen . Introduction to Quadratic Forms over Fields . 67 . . American Mathematical Society . 2005 . 0-8218-1095-2 . 1068.11023 . 2104929 .
• Book: Rajwade, A. R. . Squares . 171 . London Mathematical Society Lecture Note Series . . 1993 . 0-521-42668-5 . 0785.11022 .

Notes and References

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. 396
7. Lam (2005) p. 395