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12
1.Basicalgebraicpropertiesofintegers
4.BasicalgebraicpropertiesofZ
TheringZisaEuclideanringwithrespecttothenormfunctiondefinedas
theabsolutevalue.ItfollowsthatZisaprincipalidealdomain(PID),i.e.
adomainsuchthateveryidealisgeneratedbyoneelement.Henceevery
idealinZiscomposedofintegermultiplesnm,n∈Z,ofafixednatural
numberm.Theidealwillbedenotedby(m).
SinceeveryPIDisauniquefactorizationdomain(UFD),weseethat
ZisaUFD.Letusrecallthatauniquefactorizationdomainisadomain
suchthateverynonzerononunitcanbeexpressedasaproductofirreducible
elementsandsuchafactorizationisuniqueuptotheorderoffactorsand
multiplicationoffactorsbyunits.
TheirreducibleelementsinZaretheprimenumbersandtheirnega-
tives,hencethebasictheoremofthearithmeticofnaturalnumbersfollows
immediatelyfromtheUFDpropertyofZ.
IneveryUFDringR,foreverypair(a,b)ofnonzeroelements,onedefines
thegreatestcommondivisorgcd(a,b)asadivisorofbothelementswhichis
divisiblebyallcommondivisorsofthoseelements.Thengcd(a,b)existsand
isuniquelydetermineduptomultiplicationbyaninvertibleelement.This
notioncanbegeneralizedtoanyfinitesystemofnonzeroelements.
Recallalsothattwoelementsaresaidtoberelativelyprimeiftheir
greatestcommondivisoris1.
InthecaseofZ,thegreatestcommondivisorcanbe(uniquely)defined
asthegreatest(withrespectto≥)commondivisor.
IfthedomainRisaPID(asisZ),thenthegreatestcommondivisorof
anysetofelementsisageneratoroftheidealgeneratedbytheseelements.
Inparticular,ifelementsa,barerelativelyprime,thentheidealcontains1,
andhencethereexistx,y∈Rsuchthatax+by=1.
Next,thepropertyofbeingaUFDimpliesthatthedomainisaninte-
grallycloseddomain(ICD).RecallthatadomainRissaidtobeintegrally
closedifeveryelementfromthefieldoffractionsQ(R)whichisarootofa
monicpolynomialfromR[x]belongstoR.(Apolynomialf∈R[x]iscalled
monicifthecoefficientofthehighestpowerofxis1.)
StudyingthepropertiesofintegersandoftheirringZisthebasicaim
ofnumbertheory.However,inthedevelopmentofthetheoryithasbecome
clearthatstudyingthepropertiesofsomeotherringsconnectedinsome
wayswithZisofgreatimportance.Nowadays,thestudyofthoseringsis
consideredanimportantpartofnumbertheory,eveniftheirdirectrelation
tothepropertiesofintegersisnotvisible.InChapters3,4and5weare
goingtodescribeimportantexamplesofsuchrings(theringofGaussian
integers,theringsZmform∈N,theringsOpofp-adicintegers)andshow
howconsideringtheseringshelpsinsolvingsomeproblemsconcerningZ.In
thesecondpartofthebook,weshallshowhowtheseexamplesofringsare
placedinafarmoregeneralanddeeptheory.
