AlgebraicNumber Theory
Lecture 9
Last lecture

3rd DecemberÖDedekind domain K Fra ofractional ideal fingen 6 submodule

ideal group TriOE

ideal class group

Vian2021of k

Otosetof allfract ideals
XE KI XO EOSubgroup ofTypClk

principal ideals
adimend vector spacer Completemen
airfundamental mesh

XY

wwwDefinition 5.6

Let X

centrallysymmetric if1

convexXml XiinX is called

Vi

X EX

if for allexEXIYEHOEBCX
0not centrallysum

ab askcentrallyStattein inaep.TV

I XiERWe now state Minkowski's lattice point theorem

Theorem 5.7

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

centrally symmetricSuppose that2 vollst

vollThen X contains at leastaer with

8ooovoll

implies

Ex g n Extrito
Then there exist x x e X with
X TG

ierERZICE

We will showthat for fit

Gone nonzero lattice

VON 4point get
Proofof V

EXIL

ri.EE

S

Assuming thatEXEr vollrollt voll EInsinceg are pairwise

JEPEX garendisjoint gives

Exalso pairwise disjoint

E voll d nixAperSince

E gisVvoll Evollwhichcoverafis complete

L vollcontradiction i e thenot pairwise disjoint

we getKtg areI

6

Minkowski Theory

From nowRedk is again

onLetK Oand O a primitive

nelement i e K 0107 and the
Minimal polynomial of O is

IT

Polt
Any
Each

Oieke

EAX

Gcanbe written asdeka

dota

Dfor ihn inan O

Otgives

Y ni K

anembeddingh l

Ega ein

Somea number fieldILa0of the O are real and ei K CR

These embeddings 7arecalled realembeddings

The other embeddings come in pairs of complex

FEARwithIcici

The corresponding embeddings ei earecalled complex

conjugateshOiVenembedding

25 Ofpairs
complex embeddings

Feat embedding

Example

QIKX

PolX

2GESTÜTE

EIN AH Q

We want to embedd K into some euclidean
vector space in order to use the results onlattices

First consider all embeddings ei kj

KaKKa

aital

can be equipped withxySum overIIE

Cyisthe hermitian scalar product

E XY2linearZonzugenerates

which acts on

the Galois group Glaand also

Cy x andLx 70
for X OCThe complex conjugation FCarcaall embeddings

i eand define

Homalk El byEKICHEThis gives an involution FwithCKoKeZz

In the Example

E Esdaz

KaYenFIFA EI
The scalar product satisfies
FCN FLY

ICHDefine theTr
We havelinear

Ke
Helen Exe

Fo Tr

Trot

Fra KIK

gives themap

traceof 4k

andProposition 3.6

Fefinition 6.1
F

Let Kp denote theinvariant subspace

of ke

i e

In the example Kp

E IGERZEE

The restriction ofisKR

ZEscalar productKel ZEEonKr KR RG
R vector space Kpon theFC XDFCH FCHKp gives asince

LAND E R

4x
The euclidean vector space Kpi

is called

Minkowski spaceis called the cicalmetric and4
the associated measure Definition 5 5
iscalled canonicalmeasurefor XC Kp its volume by vol

We denoteI

Denote the real embeddings of k byk
R and the complex ones byGfn6nä

6,6TKpEIZe 1
Zo75

KG

ThenEkel Zeit R ZEE

TR pihn2sinZeRe zbIm Za

this isomorphism transforms the canonical metricinto the scalar productNoticexyde XYon2YandexaR

Therefore the canonical measure and theLebesgue

measureVol X

differ by a factor of 2
2 values XTolume

defined viaisom

Proposition 6.2
thenfjoe

Ifa

tois an ideal of On

is a complete lattice in Kp

Its fundamental mesh has volume

Talonvoll

Prof Let an an be a 2

basis of ol

22 54

5

Now for embeddingsATMdloi dla.in

aanK

Einen

E

Set

Then we havedetlptj

jaYh

Since

KikiwejagetEleken

HATvollrkIdetkikilijkulil
IdeHAlllOK.oenthenNowwepoint theorem toTheorem 6.3

Okand letATa

Minkowski's

usecanlatticeobtain the following

Letoeto bean ideal of
Ce o be real numbers forsuche c Homalk d such that GoSalon
EI

Then there exists afor all
K diEIaduEdelila

CeEE

Homolkaeiathinabi

a bin a bi

detli.itzE 42FIFCwithatoZu

OKG andH2

EGG

S lje0

The set X

GEKI Zr

centrally symmetric andConsider the isomorphism

convex

ProoffTRKR
Zz

Xewith Xe Ze and Xo Retz
ThenXp Im za

WEIR KekseFCAwhich gives

Vol X

er

Irdnebensaelf

cynicalL

RepTIce

Tefgeachpair

By CH gndfrop6nl

vonEIcweget

ETTENHEIMis2voll

Theorem 5.7 Minkowski lattice pointtha
then implies that there exist a nonzero
denwithatoila EX i e12cal

Cefor allee

Homalk c