Number Theory: An Introduction to Mathematics

322 VII The Arithmetic of Quadratic Forms

4 Supplements

It was shown in the proof of Proposition 41 that if an integer can be represented as a
sum of 3 squares of rational numbers, then it can be represented as a sum of 3 squares
of integers. A similar argument was used by Cassels (1964) to show that if a poly-
nomial can be represented as a sum ofnsquares of rational functions, then it can be
represented as a sum ofnsquares of polynomials. This was immediately generalized
by Pfister (1965) in the following way:

Proposition 43For any field F , if there exist scalarsα 1 ,...,αn∈ F and rational
functions r 1 (t),...,rn(t)∈F(t)such that

p(t)=α 1 r 1 (t)^2 +···+αnrn(t)^2

is a polynomial, then there exist polynomials p 1 (t),...,pn(t)∈F[t]such that

p(t)=α 1 p 1 (t)^2 +···+αnpn(t)^2.

Proof Suppose first thatn=1. We can writer 1 (t)=p 1 (t)/q 1 (t),wherep 1 (t)and
q 1 (t)are relatively prime polynomials andq 1 (t)has leading coefficient 1. Since

p(t)q 1 (t)^2 =α 1 p 1 (t)^2 ,

we must actually haveq 1 (t)=1.
Suppose now thatn>1 and the result holds for all smaller values ofn.Wemay
assume thatαj =0forallj, since otherwise the result follows from the induction
hypothesis. Suppose first that the quadratic form

φ=α 1 ξ 12 +···+αnξn^2

is isotropic over F. In this case there exists an invertible linear transformation
ξj=

∑n

k= 1 τjkηkwithτjk∈F(^1 ≤j,k≤n)such that

φ=η^21 −η^22 +β 3 η^23 +···+βnη^2 n,

whereβj∈Ffor allj>2. If we substitute

η 1 ={p(t)+ 1 }/ 2 ,η 2 ={p(t)− 1 }/ 2 ,ηj=0forallj> 2 ,

we obtain a representation forp(t) of the required form.
Thus we now suppose thatφis anisotropic overF. This implies thatφis also
anisotropic overF(t), since otherwise there would exist a nontrivial representation

α 1 q 1 (t)^2 +···+αnqn(t)^2 = 0 ,

whereqj(t)∈F[t]( 1 ≤j≤n), and by considering the terms of highest degree we
would obtain a contradiction.
By hypothesis there exists a representation

p(t)=α 1 {f 1 (t)/f 0 (t)}^2 +···+αn{fn(t)/f 0 (t)}^2 ,