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
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