Number Theory: An Introduction to Mathematics

318 VII The Arithmetic of Quadratic Forms

F=K(t)of rational functions in one variable with coefficients from a fieldK,the
weak Hasse principle always holds, and the strong Hasse principle holds forK=R,
but not for all fieldsK.
The strong Hasse principle also fails for polynomial forms overQof degree>2.
For example, Selmer (1951) has shown that the cubic equation 3x^3 + 4 y^3 + 5 z^3 = 0
has no nontrivial solutions inQ, although it has nontrivial solutions in every comple-
tionQv.However,Gusi ́c (1995) has proved the weak Hasse principle for non-singular
ternary cubic forms.
Finally, we draw attention to a remarkable local-global principle of Rumely (1986)
for algebraic integer solutions of arbitrary systems of polynomial equations

f 1 (ξ 1 ,...,ξn)=···=fr(ξ 1 ,...,ξn)= 0

with rational coefficients.
We now give some applications of the results which have been established.

Proposition 41A positive integer can be represented as the sum of the squares of
three integers if and only if it is not of the form 4 nb, where n≥ 0 and b≡7mod8.

Proof The necessity of the condition is easily established. Since the square of any
integer is congruent to 0,1 or 4 mod 8, the sum of three squares cannot be congruent to

7. For the same reason, if there exist integersx,y,zsuch thatx^2 +y^2 +z^2 = 4 nb,where
   n≥1andbis odd, thenx,y,zmust all be even and thus(x/ 2 )^2 +(y/ 2 )^2 +(z/ 2 )^2 =
   4 n−^1 b. By repeating the argumentntimes, we see that there is no such representation
   ifb≡7 mod 8.
   We show next that any positive integer which satisfies this necessary condition is
   the sum of three squares ofrationalnumbers. We need only show that any positive
   integera≡7 mod 8, which is not divisible by 4, is represented inQby the quadratic
   form

f=ξ 12 +ξ 22 +ξ 32.

For every odd primep, fis isotropic inQp, by Corollary 26, and hence any integer
is represented inQpbyf, by Proposition 5. By Corollary 38, it only remains to show
thatfrepresentsainQ 2.
It is easily seen that ifa≡ 1 ,3 or 5 mod 8, then there exist integersx 1 ,x 2 ,x 3 ∈
{ 0 , 1 , 2 }such that

x^21 +x 22 +x 32 ≡amod 8.

Hencea−^1 (x^21 +x 22 +x 32 )is a square inQ× 2 and frepresentsainQ 2.
Again, ifa≡2or6mod8,thena≡ 2 , 6 ,10 or 14 mod 2^4 and it is easily seen that
there exist integersx 1 ,x 2 ,x 3 ∈{ 0 , 1 , 2 , 3 }such that

x^21 +x^22 +x^23 ≡amod 2^4.

Hencea−^1 (x^21 +x 22 +x 32 )is a square inQ× 2 and frepresentsainQ 2.
To complete the proof of the proposition we show, by an elegant argument due to
Aubry (1912), that iffrepresentscinQthen it also representscinZ.