Pythagoras number

Examples

• Every positive real is a square, so p = 1.
• For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, 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 = 4.

  Properties

• Every positive integer occurs as the Pythagoras number of some formally real field.
• The Pythagoras number is related to the Stufe by p ≤ s + 1. If F is not formally real then s ≤ p ≤ s + 1, and both cases are possible: for F = C we have s = p = 1, whereas for F = F₅ we have s = 1, p = 2.
• The Pythagoras number is related to the height of a field F: if F is formally real then h is the smallest power of 2 which is not less than p; if F is not formally real then h = 2s. 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.