Number Theory: An Introduction to Mathematics

312 VII The Arithmetic of Quadratic Forms

where(a,ad)F = (b,bd)F,thenf is equivalent tog. The hypothesis implies
(−d,a)F=(−d,b)Fand hence(−d,ab)F=1. Thusξ 12 +dξ 22 representsaband
frepresentsb.Sincedetfand detgare in the same square class, it follows thatfis
equivalent tog.
Suppose next thatn ≥3 and the result holds for all smaller values ofn.Let
f(ξ 1 ,...,ξn)andg(η 1 ,...,ηn)be non-singular quadratic forms with detf=detg=
dandsF(f)=sF(g). By Proposition 31, the quadratic form

h(ξ 1 ,...,ξn,η 1 ,...,ηn)=f(ξ 1 ,...,ξn)−g(η 1 ,...,ηn)

is isotropic and hence, by Proposition 7, there exists somea 1 ∈F×which is repre-
sented by bothfandg. Thus

f∼a 1 ξ 12 +f∗, g∼a 1 η^21 +g∗,

where

f∗=a 2 ξ 22 +···+anξn^2 , g∗=b 2 η^22 +···+bnηn^2.

Evidently detf∗and detg∗are in the same square class andsF(f) =csF(f∗),
sF(g)=c′sF(g∗),where

c=(a 1 ,a 2 ···an)F=(a 1 ,a 1 )F(a 1 ,d)F=(a 1 ,b 2 ···bn)F=c′.

HencesF(f∗)=sF(g∗). It follows from the induction hypothesis thatf∗∼g∗,and
sof∼g.

3 TheHasse–MinkowskiTheorem...............................

Leta,b,cbe nonzero squarefree integers which are relatively prime in pairs. It was
proved by Legendre (1785) that the equation

ax^2 +by^2 +cz^2 = 0

has a nontrivial solution in integersx,y,zif and only ifa,b,care not all of the same
sign and the congruences

u^2 ≡−bcmoda,v^2 ≡−camodb,w^2 ≡−abmodc

are all soluble.
It was first completely proved by Gauss (1801) that every positive integer which is
not of the form 4n( 8 k+ 7 )can be represented as a sum of three squares. Legendre had
given a proof, based on the assumption that ifaandmare relatively prime positive
integers, then the arithmetic progression

a,a+m,a+ 2 m,...

contains infinitely many primes. Although his proof of this assumption was faulty,
his intuition that it had a role to play in the arithmetic theory of quadratic forms