6 1. A Brief Classical Introduction

elements of F over R; moreover, for every a G F represented by these forms,

there is a bisection between the subsets

q~1(a)

and

h~l(a)

of

Rn.

We now have two fundamental problems for quadratic forms over R:

Determine a computationally effective set of necessary and sufficient condi-

tions for two given forms to be equivalent or for one to represent the other.

Incidentally, notice that the question of whether a given form represents a

given element amounts to a special case of the problem of whether one form

represents another: a — q(x\,..., xn) if and only if

ax2

— q.

The equivalence problem for quadratic forms over R can be stated as

a matrix question: Given symmetric matrices A, C G M

n

(F), is there a

matrix T G GLn(R) such that C =

lTATl

If the answer is affirmative we

write A = C and say that A and C are congruent over R. (As before, if R

is understood we usually write "A = C"\) The representation problem for

quadratic forms has a similar matrix formulation, except that T no longer

is required to be unimodular or square.

Example 1.10. Let q = x\ + x\ and h = 2x\ +

2x2.

Then h(x) = q(Tx),

where T = ( j _ j J. Since T G M2(Z) n GL2(Q), it follows that /i —• g

and h = q. But it is NOT the case that q — h, since q represents 1 over Z

Q Z

while h does not, and hence h ^ q. In terms of matrix congruence,

z

1 0 \ / 2 0 \ . / 1 1

0 1 h o 2

V i a T="

1 - 1

while

1 0 \ / 2 0

0 1 / f I 0 2

Notice that iiq^h, say C =

lTAT

with T G GLn(R), then

|C| = \A\ •

|T|2

G |A| •

U2

G

F*/C/2

U {0}.

Let us dissect this latter expression. Here F* = F — {0} is a multiplicative

group, U — R* is the group of units of i?, and

U2

is its subgroup of squares.

So

F*/U2

is a multiplicative group, and

F*/U2

U {0} then becomes a semi-

group if we define 0 • x = 0 for all x. The discriminant of q is the element

dq = \A\ •

U2

G

F*/U2

U {0}. It follows that if q ^ ft then dq = d/i; that

is, the discriminant is a class invariant of quadratic forms. (We will have

more to say about discriminants—including some variations—later on. See

2.6 and 2.48.)

Example 1.11. If R = Z then U = {±1}, hence

U2

= {1} and so \A\ is

a class invariant, often called the determinant of the form. In practice