Algebraic number theory ANT
BasicsS I Introduction
ANTThe study offgebraicnumberfields

QI WhatAlisfield extensions

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of finite degree

iaxbia.be2dimqk

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ring of integers
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Unique AFD
2 3 factorization

p prime

FC

LEK

6 23

TED

CHAI Airreducible

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Q2WhyAZ

Alg number fields appear naturally whenstudying Diophantineequations
polynomial equations where oneisinteger rational

interested inSolutions
See ARI Chapter 17 for details

ExampleiChapter

asolution iff

KK

Linear Diophantine equation Given abwith
axt by c
Xy e ZHasChecke isacelfindmultipleof gedlaib
Ii

Pell's equationGiven

nonsquare intesernnyLagrange 1768

For any

non squaren

this has

infinitely many distinct solls x.AZiii

Sumof four squares Forabietetnne

N find akadez

withALagrange 11770 A Has
2021

02

asolution for164422

12any n1418 204362
How many

labiad ER I
Jacob 18341 If Khthenrun82m3

m divides nminbut m is notdivisible by t

Easy to prove byusing modular forms for Polt
inAb

Sum of two squares

always have

Does notna solutioneg n

Section I Ir

Fermat's

last theorem

ForngivenI

considerEyFor
Fermats claim
there are nosolutions

FL
nel trivial

Fez Pythagoras
triplesn3

nontrivial

i e X Y Ed

32 42 52
5 122 132

Hwy

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Pythasgroße
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RichardEine
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idealnumbers

AlbertTat

Okt idealsZahlbericht

FIT forAndrew
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FLT

foalrespinesgModern ANT

1 1 Primesof two squares

sumaspossible
5 12 2 7 4,11
2 12 12 3 not
17 12 42 19 X 23 X 29 22 52

It
sincesomeringtheX

13 2432

Any square is congruent to Oort modulo 4Km

4m2Mt

4m 4Mt ElETimFermat

If p 3isa

DabJabez

0modt
modt

prime thenPE I

mody

To prove this we will leave the ring of integers
and go into a larger ringDefinitioniisRA ringset R together withatwobinary operations
such that

additionandmalt

is an abelian group

Ri

70 ER

Ris

tassociatiinesab.baasemigroupassociative

Distributive property
a b t a ca be cb caatb c
is commutative ve call R commutativeIf

we just write RifRi
clear from context

Instead ofare

1

A ring RaIii

I ER
Cie

unita if there exists

iswith I aa lRi

monoidisa

A ring homomorphism isbetween

andtworing.RSaaKack

map y Rsuch thatSglatte b

Hat b
gla b

celal

9lb

If Ris are unitary one usually considers unitaryring homomorphism which satisfy cell

Alnonzerdcommutativering Ris calledaanljI ifforallabeRa.b0 implies a 0orb Oifeng.hr

The rings we consider in this course
and unitarycommutative

if not stated

Etappen R C
iiswe5Ö

atbila.be 2Kipintegral domains

Gaussianinteger

The ring ofIn RCotherwise

is not a domain since 2.3

Zu

1

are

aredomain

acanxxiNEwrite

aspxiyTro prove Ihm I I we therefore want to understandwhen a prime p can be factored like this in ZitIn I we can factor numbers almost uniquely
into primes
26 2 3uptoERunits

Definition i An element a R in anlanitalring
R is calleda baunit if there exists a be R withalof ante ofThe setsring Rais denoted

RCheckIf

1

Risa

group

RE RI 03 and Ris commutative
then R is called afieldÄH
germanExamthiB1

üIf R isadomainthen217

1 1 I iAwKERIDER

if R is not adomain then there
Might be moreunitse g R FR HEXE IKit

Definition
An element ae R with acalled irreducible if fromitwithbicep

abeorERwith PER

isobtain be R

weAn element PE RKO

ii

R is

called prime and wheneverpl abTthen pla or pl b

AcAWIThe

Inanp.ca b

domain every prime element is irreducible

converse is not always true

RRAD isaR and2 is irreducible in6

2.3

bat

2X

domainCHILIHP

does not divideandA
2X

1As

2 is not prime