Number Theory: An Introduction to Mathematics

310 VII The Arithmetic of Quadratic Forms

then(−a,−b)F =− 1 =(− 1 ,−b)Fand(−a,b)F = 1 = (− 1 ,b)F. Thus, for
allc∈F×,(−a,c)F=(− 1 ,c)Fand hence(a,c)F=1. Thereforeais a square and
the Hilbert fieldFis of type (A).

Proposition 31If F is a Hilbert field of type (B), then any quadratic form f in more
than 4 variables is isotropic.
For any prime p, the fieldQpof p-adic numbers is a Hilbert field of type (B).

Proof The quadratic formfis equivalent to a diagonal forma 1 ξ 12 +···+anξn^2 ,where

n>4. Ifg=a 1 ξ 12 +···+a 4 ξ 42 is isotropic, then so also isf.Ifgis anisotropic then,
sinceFis of type (B), it is universal and represents−a 5. This proves the first part of
the proposition.
We already know thatQpis a Hilbert field and we have already shown, after the
proof of Corollary VI.17, thatQpis not an ordered field. HenceQpis a Hilbert field of
type (B).

Proposition 10 shows that two non-singular quadratic forms innvariables, with
coefficients from a Hilbert field of type (A), are equivalent overFif and only if they
have the same positive index. We consider next the equivalence of quadratic forms
with coefficients from a Hilbert field of type (B). We will show that they are classified
by their determinant and their Hasse invariant.
If a non-singular quadratic form f, with coefficients from a Hilbert fieldF,is
equivalent to a diagonal forma 1 ξ 12 +···+anξn^2 , then itsHasse invariantis defined to
be the product of Hilbert symbols

sF(f)=

∏

1 ≤j<k≤n

(aj,ak)F.

We writesp(f)forsF(f)whenF=Qp. (It should be noted that some authors define
the Hasse invariant with

∏

j≤kin place of

∏

j<k). It must first be shown that this is
indeed an invariant off, and for this we make use ofWitt’s chain equivalence theorem:

Lemma 32Let V be a non-singular quadratic space over an arbitrary field F. If
B={u 1 ,...,un}andB′={u′ 1 ,...,u′n}are both orthogonal bases of V , then there
exists a chain of orthogonal basesB 0 ,B 1 ,...,Bm, withB 0 =BandBm=B′,
such thatBj− 1 andBjdiffer by at most 2 vectors for each j∈{ 1 ,...,m}.

Proof Since there is nothing to prove if dimV=n≤2, we assume thatn≥3and
the result holds for all smaller values ofn.Letp=p(B)be the number of nonzero
coefficients in the representation ofu′ 1 as a linear combination ofu 1 ,...,un. Without
loss of generality we may suppose

u′ 1 =

∑p

j= 1

ajuj,

whereaj= 0 ( 1 ≤j≤p).Ifp=1, we may replaceu 1 byu′ 1 and the result now
follows by applying the induction hypothesis to the subspace of all vectors orthogonal
tou′ 1. Thus we now assumep≥2. We have