üLast lecture

Theorem 4.4 Every idealaFÜ of aDedekind domain O admits a factorization
dPPr

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

Lemma 4.6andGELet pe O be asetp

XE

KI XPm.ua

prime ideal

0aEaXaI die a Xi Ef

In particular pEpp
PEppmaximalO

GaofFra ofractionali of k

Definition 4.7 DAisK

Dedekind domainOfinitely generated

submodule OtokThe fractional ideals in Gare calledintegral ideals of k1ForIia Ekaabisa fractional

called fractionalprinci alideal
We define products sums similar to ideals

ideal

Roitint 8

A G submodule otoof

K is a fractional ideal if and only if there
exists aProofe0let

CFOwith

oe be generated

hin du Then we canthe product of the denominators of aIfideala oecof Ochoose

Oi ecocobyto bearthen coe

is an

it is fingeneratedsinceare

O is noetherian

IffBr

the generators ofcoe thenäßrdBungenerate

The fractional ideals

Proposition 4.9form an abelian group the idealgron

InGfKand

theThe identity isinverseÖProofof a fractional ideal a isXE KI XO E 0Associativity
arecommutativityclear

For prime ideals

thatOlpc Oand alla

we already saw

post 0For an integral ideal Ol P
Pr
the inverse is given by b pi p
Since

ab

O implies balandifXEofoxdb which gives

Kolbe
wob

wehaveXEbö

If oe is fractional i e Corso
thenfor some CEO doforEdoedt O

CorollaryIIadmitsaEvery fractional ideal orunique representation asPp

ProofproductIT proddeWithaideal

ipod E Z and plato for almost all p
For somean Integral

Thm.MN

CEOideal

Cto
Coe iswhich has a prime decomposition

Butcodaci eanyfractional ideal is the quotient of two
integral ideals

RemarktiA

some properties of the exponents up

rplodtvpler

Vploeb1ME0Upla 20 Hp

1a Eb

Upload up b

relax

itb

min

Kp

plo up b

Definition we denote the subgroup ofIn generated by all fractional
principal ideals

1Theaquotient groupenTYP

ab with ask

Iscalled thelidealtagrafK

RemarktA Dedekind domain O isiaPID

if Clr is trivial
A Dedekind domain is a UFDiff it

1isaPID

We have the exact sequence

1GI

4KamTumbersCentTu
laIdealsmeasures thecontraction
going fromymeasures theexpansion going
fran numbers toideals

numbers to

idealsGoal

UnderstandÖ

Number fields

Clk in more detailfor

5Lattices

In the proof of Prop 1 B
we8

Ii

is factorial

already saw that lattices can help tounderstand algebraic objects

Definition 5.1
iLet Ube an n dimensional R vectorspace

A lattice in VZu tT

isa subgroup

Zumwith linearly independent kThe fai

imof theform

Um EV

is called abasis

Xiv

Xie R O ExXmumand thesefundamentalen of the lattice
Tiscalled complete ifmen1

Example D 2C2

1

is24 isaa

non complete

completelatticelatticeÄV

R vectorspacen dim

Definition 5.2

A subgroup GEV

is called

discretenbgrafifad geb arewith respect to theisolated points in V
topology

V obtains from isom VER

Every lattice f Zu

Zum C Vis a discrete subgroup Since if n m m
is a basis of V then each point
hasGawithamVm E TJA V tXivanXi ERUrn T83

neighbourhood

19 X I for it

imIndeed

the converse is also true

Propositions A subgroup To vis
alattice if and only if it is discrete

Proof

Neukirch Prop

4.2

Lemmast A lattice

ATVis complete

if and only if there exists abounded subset
MCV whose translates cover V i eUNDVH

JET
Proof

clear by taking M

Let Vo be the subspace spanned by f
Want to show Vo V
write for any nn v

with an EMFor anyAn 8h
and the Fve Vsincewe canThenVthanIgn

i efinstantlinton
nofightingM
ITbounded

Definition 5.5iVoSince Vois closed

Ieuclidean vector spaceA

finite dimensional R vector space V equippedwith a symmetric positive definite bilinear form

isaRV XViOn V we have a notion of volume Let
orthonormal basis of V For
ein en be a

ein

inder

vn withVVi
weI

dieI

define the volume of the parallelepipeduvollvn

SXNt tXnVn XiEIR OEXic1

Idetlaiil

Ideal

Sincedir dieViii

la leAA is1

detkvi.pt

we getvoll

Forlattice IaVol ärollt

Thisdefine

weT
fundamental
mesh off

independent of the choice ofisbasis of T

the change of basis
matrices hare determinant IIaDefinition 5.6
isinceLet X

V

centrallysymmetric ifconvexxex
if for allli

X is called
XEX sEXtotal

not centrally sum

Icentallückex

OStattWe now state Minkowski

Theorem 5.7

lattice point theorem

Lets be a complete latticein the euclidean vector space Vaud X aconvex subset

centrally symmetricSuppose thatThen X contains at least

one nonzero latticeao

We will showthat forfite Twitter

8oovollimplies

Ex g n Extrito
Then there exist x x e X withX TGierERZICEVON 4point getGd2 vollst

vollProofof VEXIL

ri.EE

SAssuming thatEXEr vollrollt voll E

Insinceg are pairwise

JEPEX garendisjoint gives

Exalso pairwise disjoint

E voll d nix

AperSince

E gisVvoll Evoll
which

cover

afis complete

L vollcontradiction i e thenot pairwise disjoint

we getKtg areI