324 VII The Arithmetic of Quadratic Forms
where f 1 (x),...,fs(x) ∈ R(x). The question was answered affirmatively by
Artin (1927). Artin’s solution allowed the numbersof squares to depend on the
function f, and left open the possibility that there might be no uniform bound.
Pfister showed that one can always takes= 2 n.
Finally we mention a conjecture of Oppenheim (1929–1953), that iff(ξ 1 ,...,ξn)
is a non-singular isotropic real quadratic form inn≥3 variables, which is not a scalar
multiple of a rational quadratic form, thenf(Zn)is dense inR, i.e. for eachα∈Rand
ε>0thereexistz 1 ,...,zn∈Zsuch that|f(z 1 ,...,zn)−α|<ε. (It is not difficult
to show that this is not always true forn=2.) Raghunathan (1980) made a general
conjecture about Lie groups, which he observed would imply Oppenheim’s conjec-
ture. Oppenheim’s conjecture was then proved in this way by Margulis (1987), using
deep results from the theory of Lie groups and ergodic theory. The full conjecture of
Raghunathan has now also been proved by Ratner (1991).
5 FurtherRemarks
Lam [18] gives a good introduction to the arithmetic theory of quadratic spaces. The
Hasse–Minkowski theorem is also proved in Serre [29]. Additional information is
contained in the books of Cassels [4], Kitaoka [16], Milnor and Husemoller [20],
O’Meara [22] and Scharlau [28].
Quadratic spaces were introduced (under the name ‘metric spaces’) by Witt [32].
This noteworthy paper also made several other contributions: Witt’s cancellation theo-
rem, the Witt ring, Witt’s chain equivalence theorem and the Hasse invariant in its most
general form (as described below). Quadratic spaces are treated not only in books on
the arithmetic of quadratic forms, but also in works of a purely algebraic nature, such
as Artin [1], Dieudonn ́e [8] and Jacobson [15].
An important property of the Witt ring was established by Merkur’ev (1981). In
one formulation it says that every element of order 2 in theBrauer groupof a field
Fis represented by the Clifford algebra of some quadratic form overF. For a clear
account, see Lewis [19].
Our discussion of Hilbert fields is based on Fr ̈ohlich [9]. It may be shown that any
locally compact non-archimedean valued field is a Hilbert field. Fr ̈ohlich gives other
examples, but rightly remarks that the notion of Hilbert field clarifies the structure of
the theory, even if one is interested only in thep-adic case. (The name ‘Hilbert field’
is also given to fields for which Hilbert’s irreducibility theorem is valid.)
In the study of quadratic forms over an arbitrary field F, the Hilbert symbol
(a,b/F) is a generalized quaternion algebra (more strictly, an equivalence class of
such algebras) and the Hasse invariant is a tensor product of Hilbert symbols. See, for
example, Lam [18].
Hasse’s original proof of the Hasse–Minkowski theorem is reproduced in
Hasse [13]. In principle it is the same as that given here, using a reduction argument
due to Lagrange forn=3 and Dirichlet’s theorem on primes in an arithmetic progres-
sion forn≥4.
The book of Cassels contains a proof of Theorem 36 which does not use
Dirichlet’s theorem, but it uses intricate results on genera of quadratic forms and is