theorems on sums of squares

Theorem ().

Let F be a field with characteristicPlanetmathPlanetmath not 2. The sum of squares identityPlanetmathPlanetmathPlanetmathPlanetmath of the form


where each zk is bilinearPlanetmathPlanetmath over xi and yj (with coefficientsMathworldPlanetmath in F), is possible iff n=1,2,4,8.


  1. 1.

    When the ground field is , this theorem is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the fact that the only normed real division alternative algebra is one of , , , 𝕆, as one observes that the sums of squares can be interpreted as the square of the norm defined for each of the above algebrasMathworldPlanetmathPlanetmath.

  2. 2.

    An equivalent characterization is that the above four mentioned algebras are the only real composition algebras.

A generalizationPlanetmathPlanetmath of the above is the following:

Theorem (Pfister’s Theorem).

Let F be a field of characteristic not 2. The sum of squares identity of the form


where each zk is a rational function of xi and yj (element of F(x1,,xn,y1,,yn)), is possible iff n is a power of 2.

Remark. The form of Pfister’s theorem is stated in a way so as to mirror the form of Hurwitz theorem. In fact, Pfister proved the following: if F is a field and n is a power of 2, then there exists a sum of squares identity of the form


such that each zk is a rational function of the xi and a linear function of the yj, or that

zk=j=1nrkjyj  where rkjF(x1,,xn).

Conversely, if n is not a power of 2, then there exists a field F such that the above sum of square identity does not hold for any ziF(x1,,xn,y1,,yn). Notice that zi is no longer required to be a linear function of the yj anymore.

When F is the field of reals , we have the following generalization, also due to Pfister:


If fR(X1,,Xn) is positive semidefinitePlanetmathPlanetmath, then f can be written as a sum of 2n squares.

The above theorem is very closely related to Hilbert’s 17th Problem:

Hilbert’s 17th Problem. Whether it is possible, to write a positive semidefinite rational function in n indeterminates over the reals, as a sum of squares of rational functions in n indeterminates over the reals?

The answer is yes, and it was proved by Emil Artin in 1927. Additionally, Artin showed that the answer is also yes if the reals were replaced by the rationals.


  • 1 A. Hurwitz, Über die Komposition der quadratishen Formen von beliebig vielen Variabeln, Nachrichten von der Königlichen Gesellschaft der Wissenschaften in Göttingen (1898).
  • 2 A. Pfister, Zur Darstellung definiter Funktionen als Summe von Quadraten, Inventiones Mathematicae (1967).
  • 3 A. R. Rajwade, Squares, Cambridge University Press (1993).
  • 4 J. Conway, D. A. Smith, On Quaternions and Octonions, A K Peters, LTD. (2002).
Title theorems on sums of squares
Canonical name TheoremsOnSumsOfSquares
Date of creation 2013-03-22 15:06:04
Last modified on 2013-03-22 15:06:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Theorem
Classification msc 12D15
Classification msc 16D60
Classification msc 15A63
Classification msc 11E25
Synonym Pfister theorem
Related topic MazursStructureTheorem
Related topic SumsOfTwoSquares
Related topic StufeOfAField
Related topic CayleyDicksonConstruction
Related topic CompositionAlgebra
Related topic Octonion
Defines Hurwitz theorem
Defines Pfister’s theorem
Defines Hilbert’s 17th Problem