Number Theory: An Introduction to Mathematics

320 VII The Arithmetic of Quadratic Forms

thenD:=BtBandE:=BtJBare diagonal matrices:

D=diag[d 1 ,...,dn− 1 ,n], E=diag[0,..., 0 ,n^2 ],

where dj=j(j+ 1 )for 1≤j<n. Hence, ifC=D−^1 BtAB,then

CtDC=BtAtAB.

Thus the rational matrixAsatisfies (3) if and only if the rational matrixCsatisfies

CtDC=aD+bE,

and consequently if and only if the diagonal quadratic forms

f=d 1 ξ 12 +···+dn− 1 ξn^2 − 1 +nξn^2 , g=ad 1 η^21 +···+adn− 1 η^2 n− 1 +n(a+bn)η^2 n

are equivalent overQ.
We now apply Corollary 40. Since(detg)/(detf)=an−^1 (a+bn), the condition
that detg/detfbe a square inQ×means thata+bnis a nonzero square ifnis odd and
a(a+bn)is a nonzero square ifnis even. Since ind+f=n, the condition that ind+g=
ind+fmeans thata>0anda+bn>0. The relationsp(g)=sp(f)takes the form

∏

1 ≤i<j<n

(adi,adj)p

∏

1 ≤i<n

(adi,n(a+bn))p=

∏

1 ≤i<j<n

(di,dj)p

∏

1 ≤i<n

(di,n)p.

The multiplicativity and symmetry of the Hilbert symbol imply that

(adi,adj)p=(a,a)p(a,didj)p(di,dj)p.

Since(a,a)p=(a,− 1 )p, it follows thatsp(g)=sp(f)if and only if

(a,− 1 )(pn−^1 )(n−^2 )/^2 (a,n)np−^1

∏

1 ≤i<n

(adi,a+bn)p

∏

1 ≤i<j<n

(a,didj)p= 1.

But
∏

1 ≤i<j<n

didj=(d 1 ···dn− 1 )n−^2

and, by the definition ofdj,d 1 ···dn− 1 is in the same rational square class asn. Hence
sp(g)=sp(f)if and only if

(a,− 1 )(pn−^1 )(n−^2 )/^2 (a,n)p(an,a+bn)p= 1. (4)

Ifnis odd, thena+bnis a square and (4) reduces to(a,(− 1 )(n−^1 )/^2 n)p=1.
But, sincea+bnis a square, the quadratic formaξ^2 +bnη^2 −ζ^2 is isotropic inQ
and thus(a,bn)p=1forallp. Hence(a,(− 1 )(n−^1 )/^2 n)p=1forallpif and only if
(a,(− 1 )(n−^1 )/^2 b)p=1forallp.Sincea>0, this is equivalent to (i).
Ifnis even, thena(a+bn)is a square and (4) reduces to(a,(− 1 )(n−^2 )/^2 a)p=1.
Sincea>0, this holds for allpif and only if the ternary quadratic form