Pythagoras number (original) (raw)

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Number which describes the structure of the set of squares in a given field

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 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.
• 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), Fp and the _p_-adic field Qp give (1,2); for primes p ≡ 3 (mod 4), Fp gives (2,2), and Qp 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) = 2_s_(F).[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. 396
7. ^ Lam (2005) p. 395

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 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. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.