Number Theory: An Introduction to Mathematics

6 Selected References 325

not so ‘clean’. However, Conway [6] has given an elementary approach to theequiva-
lenceof quadratic forms overQ(Proposition 39 and Corollary 40).
The book of O’Meara gives a proof of the Hasse–Minkowski theorem over any
algebraic number field which avoids Dirichlet’s theorem and is ‘cleaner’ than ours, but
it uses deep results fromclass field theory. For the latter, see Cassels and Fr ̈ohlich [5],
Garbanati [10] and Neukirch [21].
To determine if a rational quadratic form f(ξ 1 ,...,ξn) =

∑n
j,k= 1 ajkξjξk is
isotropic by means of Theorem 36 one has to show that it is isotropic in infinitely
many completions. Nevertheless, the problem is a finite one. Clearly one may assume
that the coefficientsajkare integers and, if the equationf(x 1 ,...,xn)=0 has a non-
trivial solution in rational numbers, then it also has a nontrivial solution in integers.
But Cassels has shown by elementary arguments that iff(x 1 ,...,xn)=0forsome
xj∈Z, not all zero, then thexjmay be chosen so that

max

1 ≤j≤n

|xj|≤( 3 H)(n−^1 )/^2 ,

whereH=

∑n

j,k= 1 |ajk|. See Lemma 8.1 in Chapter 6 of [4].

Williams [31] gives a sharper result for the ternary quadratic form

g(ξ,η,ζ)=aξ^2 +bη^2 +cζ^2 ,

wherea,b,care integers with greatest common divisord>0. Ifg(x,y,z)=0for
some integersx,y,z, not all zero, then these integers may be chosen so that

|x|≤|bc|^1 /^2 /d,|y|≤|ca|^1 /^2 /d,|z|≤|ab|^1 /^2 /d.

The necessity of the Bruck–Ryser–Chowla conditions for the existence of
symmetric block designs may also be established in a more elementary way, without
also proving their sufficiency for rational equivalence. See, for example, Bethet al.[2].
For the non-existence of a projective plane of order 10, see C. Lam [17].
For various manifestations of the local-global principle, see Waterhouse [30],
Hsia [14], Gusi ́c [12] and Greenet al.[11].
The work of Pfister instigated a flood of papers on the algebraic theory of quadratic
forms. The books of Lam and Scharlau give an account of these developments. For
Hilbert’s 17th problem, see also Pfister [23], [24] and Rajwade [25].
Although a positive integer which is a sum ofnrational squares is also a sum ofn
squares of integers, the same does not hold for higher powers. For example,

5906 =( 149 / 17 )^4 +( 25 / 17 )^4 ,

but there do not exist integersm,nsuch that 5906=m^4 +n^4 ,since9^4 >5906,
2 · 74 <5906 and 5906− 84 =1810 is not a fourth power. For the representation of a
polynomial as a sum of squares of polynomials, see Rudin [27].
For Oppenheim’s conjecture, see Dani and Margulis [7], Borel [3] and Ratner [26].

6 SelectedReferences

[1] E. Artin,Geometric algebra, reprinted, Wiley, New York, 1988. [Original edition, 1957]