In 1900, David Hilbert [H4]′ in his famous address at the International Conress of Mathematicians in Paris proposed as his 17th problem the following:

Hilbert's conjecture. Let f(X1, …, Xn) ∈ R(X1, …, Xn). A necessary and sufficient condition that f is a sum of squares in R(X1, …, Xn) is that f is positive definite (i.e. f(a1, …, an) 0 for all a1, …, an ∈ R for which f is defined).

A similar conjecture holds for Q(X1, …, Xn). These conjectures were proved by Artin [A6] in 1927 for both R and Q, but one still didn't know how many squares are needed for the representation. Some results were of course known when the number of variables n = 1 or 2. Let us first look at the field R.

In R(X) two squares suffice:

Theorem 4.1. Let f(X) ∈ R(X) be positive definite; then f(X) is a sum of two squares.

This had already been proved by Hilbert in 1893, as also was the next result.

Theorem 4.2 (Hilbert (1893)). Let f(X, Y) ∈ R(X, Y) be positive definite; then f(X, Y) is a sum of four squares.

This was first proved by Hilbert [H4] and later again by Witt. We shall give a proof due to Pfister [P5]. This proof has the advantage that it can be generalized to the case of n variables.

In 1966, James Ax (unpublished) proved that in R(X, Y, Z) eight squares suffice. This has now been proved by Pfister in a very elegant way (see Chapter 5).