5 The Modular Group 207denote the number of primitive positive definite quadratic forms with discriminantD
which are properly inequivalent. By what has been said,h†(D)is finite.
Consider next the indefinite case:
f=ax^2 +bxy+cy^2wherea,b,c∈RandD>0. Ifa=0, we can write
f=a(x−ξy)(x−ηy),whereξ,ηare the distinct real roots of the polynomialat^2 +bt+c. It follows from
Lemma 0 that, ifξandηare irrational, thenf is properly equivalent to a formf′
for whichξ′>1and− 1 <η′<0. Such a quadratic formf′is said to bereduced.
Evidentlyf′is reduced if and only if−f′is reduced. Thus we may supposea′>0,
and thenf′is reduced if and only if
0 <√
D+b′< 2 a′<√
D−b′.If the coefficients offare integers and the positive integerDis not a square, then
a=0andξ,ηare conjugate real quadratic irrationals. In this case, as we already
saw in Section 3, there are only finitely many reduced forms with discriminantD.For
any integerD>0 which is not a square, leth†(D)denote the number of primitive
quadratic forms with discriminantDwhich are properly inequivalent. By what has
been said,h†(D)is finite.
It should be noted that, for any quadratic form fwith integer coefficients, the
discriminantD≡0or1mod4.Moreover,foranyD≡0or1mod4,thereisa
quadratic formfwith integer coefficients and with discriminantD; for example,
f=x^2 −Dy^2 /4ifD≡0mod4,
f=x^2 +xy+( 1 −D)y^2 /4ifD≡1mod4.The preceding results for quadratic forms can also be restated in terms of quadratic
fields. By making correspond to the ideal with basisβ=a,γ=b+cωin the quadratic
fieldQ(
√
d)the binary quadratic form{ββ′x^2 +(βγ′+β′γ)xy+γγ′y^2 }/ac,one can establish a bijective map between ‘strict’ equivalence classes of ideals in
Q(
√
d)and proper equivalence classes of binary quadratic forms with discriminantD,
where
D= 4 d ifd≡2or3mod4,
D=d ifd≡1mod4.(The middle coefficientboff=ax^2 +bxy+cy^2 was not required to be even in order
to obtain this one-to-one correspondence.) Since any ideal class is either a strict ideal
class or the union of two strict ideal classes, the finiteness of the class numberh(d)of
the quadratic fieldQ(
√
d)thus follows from the finiteness ofh†(D).