Number Theory: An Introduction to Mathematics

3 The Hasse–Minkowski Theorem 319

Let

(x,y)={f(x+y)−f(x)−f(y)}/ 2

be the symmetric bilinear form associated withf,sothatf(x)=(x,x), and assume
there exists a pointx ∈Q^3 such that(x,x)=c ∈Z.Ifx ∈/Z^3 , we can choose
z∈ Z^3 so that each coordinate ofzdiffers in absolute value by at most 1/2 from
the corresponding coordinate ofx. Hence if we puty = x−z,theny =0and
0 <(y,y)≤ 3 /4.
Ifx′=x−λy,whereλ= 2 (x,y)/(y,y),thenx′∈Q^3 and(x′,x′)=(x,x)=c.
Substitutingy=x−z, we obtain

(y,y)x′=(y,y)x− 2 (x,y)y={(z,z)−(x,x)}x+ 2 {(x,x)−(x,z)}z.

Ifm>0 is the least common denominator of the coordinates ofx,sothatmx∈Z^3 ,it
follows that

m(y,y)x′={(z,z)−c)}mx+ 2 {mc−(mx,z)}z∈Z^3.

But

m(y,y)=m{(x,x)− 2 (x,z)+(z,z)}=mc− 2 (mx,z)+m(z,z)∈Z.

Thus ifm′>0 is the least common denominator of the coordinates ofx′,thenm′
dividesm(y,y). Hencem′≤( 3 / 4 )m.Ifx′∈/Z^3 , we can repeat the argument with
xreplaced byx′. After performing the process finitely many times we must obtain a
pointx∗∈Z^3 such that(x∗,x∗)=c.

As another application of the preceding results we now prove

Proposition 42Let n,a,b be integers with n> 1. Then there exists a nonsingular
n×n rational matrix A such that

AtA=aIn+bJn, (3)

where Jnis the n×n matrix with all entries 1 , if and only if a> 0 ,a+bn> 0 and

(i)for n odd: a+bn is a square and the quadratic form

aξ^2 +(− 1 )(n−^1 )/^2 bη^2 −ζ^2

is isotropic inQ;
(ii)for n even: a(a+bn)is a square and either n≡0mod4,orn≡2mod4and a
is a sum of two squares.

Proof If we put

B=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

11 ... 11

− 11 ... 11

0 − 2 ... 11

... ... ...

00 ... 1 −n 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

,