Number Theory: An Introduction to Mathematics

316 VII The Arithmetic of Quadratic Forms

By the Chinese remainder theorem (Corollary II.38), the simultaneous congruences

c≡b 2 mod 2ε^2 +^3 ,c≡bpmodpεp+^1 for every oddp∈S,

have a solutionc∈Z, that is uniquely determined modm,wherem= 4

∏

p∈Sp

εp+ (^1).
In exactly the same way as before we can replacebpbycfor all primesp∈S.By
choosingcto have the same sign asc∞, we can takecv=cfor allv∈S.
Ifd=

∏

p∈Sp

εpis the greatest common divisor ofcandmthen, by Dirichlet’s

theorem on primes in an arithmetic progression, there exists an integerkwith the same
sign ascsuch that

c/d+km/d=±q,

whereqis a prime. If we put

a=c+km=±dq,

thenqis the only prime divisor ofawhich is not inSand the quadratic forms

g∗=−aξ 02 +a 1 ξ 12 +a 2 ξ 22 , h∗=a 3 ξ 32 +···+anξn^2 +aξn^2 + 1

are isotropic inQvfor everyv∈S,sincec−^1 ais a square inQ×v.
For all primespnot inS, exceptp=q,ais not divisible byp. Hence, by the
definition ofSand Corollary 26,g∗is isotropic inQvfor allv, except possiblyv=q.
Consequently, by the final remark of part (ii) of the proof,g∗is isotropic inQ.
Suppose first thatn=4. In this case, in the same way,h∗=a 3 ξ 32 +a 4 ξ 42 +aξ 52 is
also isotropic inQ. Hence, by Proposition 6, there existy 1 ,...,y 4 ∈Qsuch that

a 1 y 12 +a 2 y 22 =a=−a 3 y 32 −a 4 y^24.

Thusfis isotropic inQ.
Suppose next thatn≥5 and the result holds for all smaller values ofn. Then the
quadratic formh∗is isotropic inQv, not only forv∈S,butforallv.Forifpis a
prime which is not inS,thena 3 ,a 4 ,a 5 are not divisible byp. It follows from Corol-
lary 26 that the quadratic forma 3 ξ 32 +a 4 ξ 42 +a 5 ξ 52 is isotropic inQp, and henceh∗is
also. Sinceh∗is a non-singular quadratic form inn−1 variables, it follows from the
induction hypothesis thath∗is isotropic inQ. The proof can now be completed in the
same way as forn=4.

Corollary 37A non-singular rational quadratic form in n≥ 5 variables is isotropic
inQif and only if it is neither positive definite nor negative definite.

Proof This follows at once from Theorem 36, on account of Propositions 10
and 31.

Corollary 38A non-singular quadratic form over the rational fieldQrepresents a
nonzero rational number c inQif and only if it represents c in every completionQv.