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Introduction
Thebookisintendedtoserveasamanualforahalf-yearintroductorycourse
ofnumbertheory.Here“introductory”meansthefirstlecturecourseinnum-
bertheorythestudentswilltake.However,wedonotassumethatanymore
advancedcourseswillfollow—onthecontrary,wesupposethat,formost
listeners,thiswillbetheonlycontactwithnumbertheoryduringtheiruni-
versitystudies.Therefore,ouraimistopresentthetheorytotheextent
whichinourviewisindispensabletoeverymathematicsstudent,regardless
oftheirfuturespecialization.However,wehopethatsomereaderswillnot
restrictthemselvestothisintroductorypart,butcontinuetheirstudyofthat
disciplineduetoitsbeauty,depthandintriguingperspectives.
Inthisbook,wehavesetsometargetpoints,importantfornumber
theoryduetothemethodsusedandessentialtothedevelopmentofthe
theory.Theseincludethelocalmethods,theQuadraticReciprocityLaw,the
PrimeNumberTheorem,uniquefactorizationofidealsinringsofalgebraic
integersinfiniteextensionsofQ,unitsintheringsofintegersinquadratic
extensionsofQ,anddecompositionofprimesinringsofintegers.Thischoice
oftargetsenablespresentingthemostimportantproblemsofnumbertheory
andtherangeofmethodsused,algebraic,analyticandgeometric.
Itiscleartotheauthorsthatalthoughtheveryrestrictedframeworkmay
leavethereaderunsatisfied,extendingthemateriale.g.bypresentingthe
detailsoftheproofsthatareonlysketchedcannotberealizedwithinaone-
semestercourse.Tosomeextent,producingthisfeelingofunsatisfactionwas
intentional:wewantedtoshowthattheresultspresentedarenotcomplete
andgivevastopportunitiesforfurtherstudyandresearch.
Thisbookisbasedonthelecturenotes(inEnglish)thatthefirstauthor
distributedamongthelistenersofhis2011–2012lecturesatWarsawUniver-
sity.Thesecondauthorcorrectedandcomplementedthesenotesinvarious
ways,andalsoaddedtheproblemsections.
Accordingtoourexperience,withsomeself-restraintofthelecturerto-
getherwithrelegatingsomepartsofthematerialtoclasses,thismaterial
canberealizedwithinaone-semestercourse(twohoursoflecturesandtwo
