Pythagoras number

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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.


  • 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.


  • 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]


  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