Pythagoras Numbers of Fields

The study of sums of squares in a ring or a field is a classic topic in algebra and number theory. In this context, several questions arise naturally. For example, which elements can be represented as sums of squares, and if an element can be written as a sum of squares, how many squares are actually needed ? For instance, for an integer n to be a sum of squares of integers, we must obviously have that n ≥ 0, and it is well known that each such n can in fact be written as a sum of four integer squares, the first published proof of this result being due to Lagrange. Furthermore, in general one needs four squares as, for example, 7 cannot be written as a sum of three integer squares. This result readily carries over to the rational numbers Q : x ∈ Q is a sum of squares of elements in Q if and only if x ≥ 0, and in this case x can be written as a sum of ≤ 4 squares. However, 7 cannot be written as a sum of three squares in Q. If we define the Pythagoras number p(R) of a ring R to be the least integer n (if such an integer exists) such that each sum of squares in R can be written as a sum of ≤ n squares, then the above shows that p(Z) = p(Q) = 4. For number fields F in general, it follows readily from the Hasse-Minkowski theorem that p(F ) ≤ 4. If R is an order in F , then p(R) is finite but can grow arbitrarily large in the case of a totally real F (cf. [Pe], [Sc]). Hilbert’s 17th problem is another classic example concerning sums of squares, the problem itself being essentially the following : Let F = R(X1, · · · , Xn) be the rational function field in n variables over the real numbers R, and let f ∈ F be a rational function such that f(a) ≥ 0 for all a = (a1, · · · , an) ∈ R where f is defined (f is then called positive semi-definite). Is f a sum of squares in F ? (It can readily be shown that in order to be a sum of squares, f must necessarily be positive semi-definite.) This problem has been given a positive answer by Artin in 1927 [A]. One can furthermore show that in this situation p(F ) ≤ 2, so that in particular p(R(X1)) = 2 (note that for example 1 + X 1 is a sum of two squares which is not a square), and for n = 2, it was shown by Cassels-Ellison-Pfister in 1971 [CEP] that p(R(X1, X2)) = 4. For n ≥ 3 the precise value is not known. For this and more on generalized versions of Hilbert’s 17th problem, we refer to the exposition given in Chapters 6 and 7 of Pfister’s beautiful book [Pf], where one also finds an account of the current state of knowledge regarding problems related to the Pythagoras number (also for rings). One question asked there was which