Pythagoras number
In this article we are going to delve into the topic of Pythagoras number, an issue that has sparked interest and debate in recent times. Pythagoras number and its implications in our society have been discussed from different areas, so it is crucial to address this issue in an exhaustive and objective manner. Along these lines, we will analyze the different aspects related to Pythagoras number, exploring its origins, evolution and repercussions in the current context. Likewise, we will stop at the different perspectives that exist around Pythagoras number, considering opinions and arguments from experts in the field. Ultimately, the objective of this article is to shed light on Pythagoras number and offer a detailed and balanced view that allows the reader to fully understand this matter and form their own judgment on it.
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 (2k,2k) or (2k,2k + 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) = 2s(F).[7]
Notes
References
- 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.