304 VII The Arithmetic of Quadratic Forms
The following lemma shows that the Hilbert symbol can also be defined in an
asymmetric way:
Lemma 22For any field F and any a,b∈F×,(a,b)F= 1 if and only if the binary
quadratic form fa=ξ^2 −aη^2 represents b. Moreover, for any a∈F×,thesetGaof
all b∈F×which are represented by fais a subgroup of F×.
Proof Suppose first thatax^2 +by^2 =1forsomex,y ∈ F.Ifais a square, the
quadratic formfais isotropic and hence universal. Ifais not a square, theny=0and
(y−^1 )^2 −a(xy−^1 )^2 =b.
Suppose next thatu^2 −av^2 =bfor someu,v∈ F.If−ba−^1 is a square, the
quadratic formaξ^2 +bη^2 is isotropic and hence universal. If−ba−^1 is not a square,
thenu=0anda(vu−^1 )^2 +b(u−^1 )^2 =1.
It is obvious that ifb∈Ga,thenalsob−^1 ∈Ga, and it is easily verified that if
ζ 1 =ξ 1 η 1 +aξ 2 η 2 ,ζ 2 =ξ 1 η 2 +ξ 2 η 1 ,then
ζ 12 −aζ 22 =(ξ 12 −aξ 22 )(η 12 −aη^22 ).(In fact this is just Brahmagupta’s identity, already encountered in§4 of Chapter IV.)
It follows thatGais a subgroup ofF×.
Proposition 23For any field F , the Hilbert symbol has the following properties:
(i)(a,b)F=(b,a)F,
(ii)(a,bc^2 )F=(a,b)Ffor any c∈F×,
(iii)(a, 1 )F= 1 ,
(iv)(a,−ab)F=(a,b)F,
(v)if(a,b)F= 1 ,then(a,bc)F=(a,c)Ffor any c∈F×.
Proof The first three properties follow immediately from the definition. The fourth
property follows from Lemma 22. For, sinceGais a group andfarepresents−a, fa
represents−abif and only if it representsb. The proof of (v) is similar: iffarepresents
b, then it representsbcif and only if it representsc.
The Hilbert symbol will now be evaluated for the real fieldR=Q∞and thep-adic
fieldsQpstudied in Chapter VI. In these cases it will be denoted simply by(a,b)∞,
resp.(a,b)p. For the real field, we obtain at once from the definition of the Hilbert
symbol
Proposition 24Let a,b∈R×.Then(a,b)∞=− 1 if and only if both a< 0 and
b< 0.
To eva l u a t e(a,b)p, we first note that we can writea=pαa′,b=pβb′,where
α,β∈Zand|a′|p=|b′|p=1. It follows from (i), (ii) of Proposition 23 that we may
assumeα,β∈{ 0 , 1 }. Furthermore, by (ii), (iv) of Proposition 23 we may assume that
αandβare not both 1. Thus we are reduced to the case whereais ap-adic unit and
eitherbis ap-adic unit orb=pb′,whereb′is ap-adic unit. To evaluate(a,b)punder
these assumptions we will use the conditions for ap-adic unit to be a square which
were derived in Chapter VI. It is convenient to treat the casep=2 separately.