Number Theory: An Introduction to Mathematics

146 III More on Divisibility

An idealA={ 0 }is said to bedivisibleby an idealB,andBis said to be afactor

ofA, if there exists an idealCsuch thatA=BC. For example,Ais divisible by itself

and byR,sinceA=AR. ThusRis an identity element for multiplication of ideals.

Now takeR=Odto be the ring of all integers of the quadratic fieldQ(

√

d).We

will show that in this case much more can be said.

Proposition 14Let A={ 0 }be an ideal inOd. Then there existβ,γ∈A such that

everyα∈A can be uniquely represented in the form

α=mβ+nγ(m,n∈Z).

Furthermore, ifωis defined as in Proposition 11, we may takeβ=a,γ=b+cω,

where a,b,c∈Z,a> 0 ,c> 0 , c divides both a and b, and ac dividesγγ′,i.e.

b^2 −dc^2 ≡0modac if d≡ 2 or3mod4,

b(b−c)−(d− 1 )c^2 / 4 ≡0modac if d≡1mod4.

Proof SinceAis an ideal, the setJof allz∈Zsuch thaty+zω∈Afor somey∈Zis
an ideal inZ. MoreoverJ={ 0 },sinceA={ 0 }andα∈Aimpliesαω∈A.SinceZis
a principal ideal domain, it follows that there existsc>0 such thatJ={nc:n∈Z}.
Sincec∈J, there existsb∈Zsuch thatγ:=b+cω∈A.
MoreoverAcontains some nonzerox∈Z,sinceα∈Aimpliesαα′∈A. Since the
setIof allx∈Z∩Ais an ideal inZ, there existsa>0 such thatI={ma:m∈Z}.
For anyα=y+zω∈Awe havez=ncfor somen∈Zandα−nγ=y−nb=ma
for somem∈Z. Thusα=mβ+nγwithβ=a. The representation is unique, since
γis irrational.
Sinceβω∈A,wehave

aω=ra+s(b+cω) for uniquer,s∈Z.

Thusa=scandra+sb=0, which together implyb=−rc.Sinceγω∈A,we
have also

(b+cω)ω=ma+n(b+cω) for uniquem,n∈Z.

Ifd≡2or3mod4,thenω^2 =d. In this casen=−r,cd=ma−rband hence
dc^2 =mac+b^2 .Ifd≡1 mod 4, thenω^2 =−ω+(d− 1 )/4. Hencen=−(r+ 1 ),
c(d− 1 )/ 4 =ma−rb−band(d− 1 )c^2 / 4 =mac+b(b−c).

IfAis an ideal inOd, then the setA′={α′:α∈A}of all conjugates of elements

ofAis also an ideal inOd. We callA′theconjugateofA.

Proposition 15If A={ 0 }is an ideal inOd,then AA′=lOdfor some l∈N.

Proof Chooseβ,γso that A ={mβ+nγ:m,n ∈ Z}.ThenAA′consists of
all integral linear combinations ofββ′,βγ′,β′γ andγγ′.Furthermorer = ββ′,
s=βγ′+β′γandt =γγ′are all inZ.Iflis the greatest common divisor of
r,sandt,thenl∈AA′,bytheB ́ezout identity, and hencelOd⊆AA′.
On the other hand,βγ′andβ′γare roots of the quadratic equation