2 1. A Brief Classical Introduction

polynomial of the form

n

q\x\,... )Xn) = y ^ ciijXiXj G ±* \x\,... ,xn\.

Because multiplication in F is commutative, in considering q(xi,..., xn)

as a function

Rn

— F without loss of generality we can (and will, from

now on) suppose that the coefficients of q satisfy the condition a^ = a^; just

replace each of these coefficients by lJ ——. (Note that here the underlying

assumption that char F ^ 2 is needed.) With this convention in force, q is

now given by the symmetric matrix A — (a^) G Mn(F). This symmetric

matrix is the Gram matrix of q.

Let a G F. We say the form q represents a over i?, denoted a —• g, if

R

there are Ai,..., An G R such that g(Ai,..., An) = a. More briefly: viewing

q as a function

Rn

— F, the statement a — q means a G range q. (A

R

more general notion of representation will come in Definition 2.8.)

An instance of the representation problem is this: Given a form g,

for which a G F is it true that a — ql

R

Example 1.2. Pythagorean triples, mentioned earlier, are precisely the

solutions for the representation 0 — x\ + x\ — x\.

Example 1.3. Brahmagupta (598-668) could generate Pythagorean triples,

and he studied representations by other quadratic forms as well. For in-

stance, from a given representation 1 —•

x2

—

ay2

(with a G N) he could

produce infinitely many.

Example 1.4. (i) Fermat (1601-1665) studied sums of integer squares,

and here is one of his typical results, stated by him in 1640 in a private

communication—along with hints on his method of proof—and with a for-

mal proof published by Euler in 1754: If n G N has prime factorization

n =

p^1

• •

-Prr,

then n —

x2

+

y2

if and only if c^ = 0 (mod 2) for all i

such that pi = 3 (mod 4).

(ii) In 1967 Leahey [Le] returned to Fermat's result, but replacing Z with

a polynomial ring. Let h(x) G Fq[x]. (Here ¥q denotes a finite field with q

elements, with q odd.) Say h(x) = ep

^1{x)

• •

-p^r(x),

with the Pi(x) prime

(monic and irreducible) polynomials and e G F*. Then h(x) is a sum of

two polynomial squares—that is, h(x) —

X2

+

Y2—if

and only if c^ = 0

¥q[x]

(mod 2) for all i such that dpi(x) = 1 (mod 2). (Here we use "9" for the

degree function.)