Number Theory: An Introduction to Mathematics

2 Quadratic Fields 147

x^2 −sx+rt= 0

with integer coefficientss=βγ′+β′γandrt=ββ′γγ′. It follows thatβγ′/land
β′γ/lare roots of the quadratic equation

y^2 −(s/l)y+rt/l^2 = 0 ,

which also has integer coefficients. Sinceβγ′/landβ′γ/lare inQ(

√

d), this means
that they are inOd. Thusβγ′andβ′γare inlOd. Since alsoββ′andγγ′are inlOd,
it follows thatAA′⊆lOd.

If in the proof of Proposition 15 we chooseβ=aandγ=b+cωas in the state-
ment of Proposition 14, then in the statement of Proposition 15 we will havel=ac.
Since the proof of this whend≡1 mod 4 is similar, we give the proof only ford≡ 2
or 3 mod 4. In this caseω=

√

dand hencer=a^2 ,s= 2 ab,t=b^2 −dc^2 .Wewish
to show thatacis the greatest common divisor ofr,sandt. Thus if we put

a=cu,b=cv,t=acw,

then we wish to show thatu, 2 v andwhave greatest common divisor 1. Since
uw=v^2 −danddis square-free, a common divisor greater than 1 can only be 2.
But if 2 were a common divisor, we would havev^2 ≡dmod 4, which is impossible,
becaused≡2or3mod4.
We can now show that multiplication of ideals satisfies the cancellation law:

Proposition 16If A,B,C are ideals inOdwith A={ 0 },then AB=AC implies
B=C.

Proof By multiplying by the conjugateA′ofAwe obtainAA′B=AA′Cand hence,
by Proposition 15,lB=lCfor some positive integerl. But this impliesB=C.

Proposition 17Let A and B be nonzero ideals inOd. Then A is divisible by B if and
only if A⊆B.

Proof IfA=BCfor some idealC,thenA⊆B, by the definition of the product of
two ideals.
Conversely, supposeA⊆B. By Proposition 15,BB′=lOdfor some positive
integerl. HenceAB′ ⊆lOd. It follows thatAB′ =lCfor some idealC. Thus
AB′=BB′Cand so, by Proposition 16,A=BC.

Corollary 18Let A and B be nonzero ideals inOd. If D is the set of all elements
a+b, with a∈A and b∈B, then D is an ideal and is a factor of both A and B.
Moreover, every common factor of A and B is also a factor of D.

Proof It follows at once from its definition thatDis an ideal. MoreoverDcon-
tains bothAandB, since 0 is an element of any ideal. Evidently also any ideal
Cwhich contains bothAand B also containsD. The result now follows from
Proposition 17.