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An basis of LIK

anLbiFi h.in

embeddings ofLinkE

disdetltruki

EK7FrackLena

KÖ Eeignung

Bijinte ÄEYintegrally

pes 0wir

WEB integral

basisofBoverAbhey

AW tanwn

an an EA

fingen B submodule M 0 oft
has integral basis over Hand u Ein

Prop 3.15

Vanir

CorollaryK

numberfield

Everyfingen

Ok module a has 2 basis

dlod dlw.in wal

dudok

discriminant of k

Proposition 3.19 If a c od are two nonzero
fin gen Ok submodules of k then theis finite and satisfies
a oindex

dlafol on dla

Use Structure theorem

Prooffingenfor modules
over PID

4 Dedekind domains
Proposition 4 I

ProofK alg Number field

In OK every non mit L O can be factored into
a product of irreducible elements
aSinceifL

ßJ

ThenZ

weis not irreducible

withnon units

haveNuala

Since Nfp Naacp

wecan write

pigeon
ezez

NflNu

AB

weget

14 NABIL

NGHI

INGA

The statement then follows inductivelyA

have seen this decomposition isunique But we will see that theAs wenotdecomposition into prime ideals is uniqueand more generally Dedekind domains

Definition 4.2

ifor On

A domain Ris called anoethetin

iit isa

it is integrally closed

Es gemein

coordinate ringaffine variety

nonsinnlar

every nonzeroprime ideal in R is maximal
Krall dimension 1

curve

coordinate

Of homeinsular rings

are Dedekindcurves

domainring of integers On ofProposition The
numberanalg

domainfield K is a Dedekind

ProofOk is noetherian since it isia finitely generated Z2 noetherian E1

O noetherianKis the integral closure of Zink
integrally closed by corollary 2.8

Oki eimodule

be a primeideal in OK

let pononyepconsider

9an 9Then p ok is a prime ideal in Z
This ideal it non zero sincefor aZerott

909JER

We see do O
otherwise this n would not be minimaland do E 2h p
ps Z lolof minimal degree

There exist a prime number p withPre
Wantto show

Gp is a fieldÖpÖ has Kas2asubring

and all XE Ökolare algebraic
XPn XBooverk ie

Ofor some B e F Assuming n is minimal
weget Boto and therefore
Xhas an1LEGIEßÖ

inversep
Theorem 4.4 Every ideal

is a fieldis maximalaA

Ö of aDedekind domain O admits a factorization
dPPr

which isinto nonzero prime ideals
p of O
unique up to the order of the factors

To

prove this we will need two Lemmas

O be a Dedekind domain

LetLemma 45

For every ideal oto of O

there exist

nonzerowithprime ideals p

Pie Pr E

off forwhich there exist noO is noetherianamaximalelemente in M with respect

prime ideal

there arebThis
otherwisebbeandbeta

Settingab

haveaossacan

not bewould holdi eO with b b e oe butolwesp withp

if Mt 0 then there

andto the inclusionad

Let M be the Set of ideals GOProofexistProatoeOer b toOl OlEdSinceolagetis anmaximal element in M weaMR

7 prime ideals

ppi.fipeoe.ae soA

Lemma 4.6
GE

Pied0M

andPiPre ofLet pe O be aprime ideal

setpXEKI XP

haveapdThen weO

for any ideo

EaiXilaieoeiXiE

Proof.IfpeldthenLetpdFKand a ep withok KaaoEmma7p

Choosep

asr

arsincep

Sincerchoosep

Ephaveweotherwise 7 dieaashall as possible

Then for one

AssumeEpP

pile for allEP

it

then we get p p

pis maximal

is minimal we have
Pb E Pzf

üb

OOn the other handapbps aaäbpsonabeff
Gp

NowO

letO beoec

noetherianoefingen

Let EdanSupposethat ap oep we have

all XEiebe generaton

Then forÖ

di diA EOwithAx

integraleroEG

apof olXLdet AX

idealanpdE0XEaidiGKGA

ProofofTheorem
M

ExisteSet of ideals la li
which do not admit a prime

idealdecomposition

If MIO then M has a maximal elementoecMoe is not

Sinceideal

O is noetheriansincewithp

Lemma 4.6

prime there exist
olOea maximal

EPPEPP O
get ppb O

Fdp

Since p is maximal we
we getand with O Ep

EppaWe

have

ap

0O

otherwise

aoepp pSince oe is maximal in M with respect to theM i einclusion we see that apaafpProepppp pMQMindy p
two factorization ofFor p prime

ab Ep

we have

9into prime ideals

adeeporbe p

Notation

plabNow weget that pplo or pl b
divides one of the9Sincei esay p 99 Ep
9 is maximal
we get 9p
P FO
Multiplying with pit gives

Pz

9PrContinuing this shows res
possibleandrenumbering

Therefore any ideal
be9s

dt d

uniquely up to reordering

oepp

p 9 afterof O

writtenascanAl and pairwise distinct
with hikideals PprimePrG

Dedekind domain

Definition 4.7
isofaAFra ofractionali of kfinitely generatedkKO

submodule Oto