AlgebraicNumber Theory
Lecture 13

K number field

21st January

real embeddings
pairs of complexembrsj KAF ZHE

GESTÜTEN
KEKEFIETE

Nutte

IIINIKaaKöe

EKTEIR

HitsR

Proposition 9.1

The sequence

Mal1ATGKa

is exact i e Kerl µ k
Lemma 9.2 Foragiven

Ok
ano

In
2 there areuptoasaaesanrunan.ea.in

NKlaKka
Proposition 9.3 The group Pi

lattice in the Crest
Hi eTER

complete

dimensional vectorspace

Theorem 9.4

Dirichlet's unit theorem

The unit group Of is given by a direct
product of the cyclic group µ k and agroup of rank rest

free abelian

Remark9i e

Of EMIN OZi

By Thm 9.4 there exist units
ItEin Etres

fundamental units
be writtencancalled the1uniteas

L EiGxEE

he µ K and k

withthat anysuchKER

The unit group Ok is finite iff rts 1 0i
FI

S O

FO

SI

ACK OftK

K Q

Md with doo squarefree
Limas quadratic fieldIn thisi

OKcase

K QU

GE

K Q A

OKFor
weII

have
Ires

OKKEv2sI

and0i e0

IR X Z

other unit

canaOf53 26QR

dohasBMy

Therefore there existanyRAZIi

real quadratic field K

arank

except forunite such thatbe writtenas

IE

E E and E also satisfy thisnproperty and among these four one usuallychooses the unique one which is positive and1

This is called

If d 2,3
therefore

the fundamental unit of K

modt weNatalsaw2

thatä b'dIPelltesuation

unit then a b'd I xcase considering small
thisInd7
eig
values for b in Bd we see that b 3

If X isgives

athe smallest solution toSince RT
viewed as378Eawith a8is the fundamental unit

ARofthe H can besubspaceof euclidean space

We can ask for the volume of theCHfundamental mesh vollof Ello

Et with tunits and set

Let E

Hta X

be fundamental

restTA Illle

R

It has length I and is orthogonal to H
The fundamental

mech

X Ei

X Et

and

we

haveof T is spanned by

VoIp KI

Volpe P

LIEDICEDHal

deff

IIIii

Eii

voten

PjFIüexcoderns Ru

whereRu is the absolute value of the determinantarbitrarymatrixlaofantheminorRyProfver lofX Et

Kati
isof rank t

YEAHcalled theregulatorofIn

add all rows to one

arbitraryrow

Every entry will vanishexcept for the entry in the first column
whichWeissawPts

FIA

the regulator appearing inTheorem 7.10

Analytic class numberformula

Theorienein HAGA

Z 1

Analytic dass number formulaZÜGE

wherewafffranitsinn
Resz Lutz

of Dedekind domains

10 Extensions
From now

weFracht

case

Ex Q

consider again the more generalfin extension

KCVU2

CHA
Dedekinddomain

CL

KnumberfielduEO
integralclosure

Assume further that chalkGoal

Exof A intoorUnderstand how prime ideals
behavein Olukepc A

Proposition 10.1i16isaDedekind domain

Every ideal of O is a finitely generatedA moduleSince O is the integral closure ofA it is integrally closed Corollary 2.8

ProofoLet anbedudiscriminant da basisof Uk withdki.mn fdet bildDbi L

Ho Prop 3.10

We canchoose

die O and by Lemma 3.13

O E AG
M isafinAFM

A module andgenby Prop 1.40 noetherian
Every ideal of O is contained in M

thereforeand therefore fin gen A moduleTherefore they are also fin gen G modules

O is noetherian

Let Pto beSetpPn Aaprime idealof O

This is a prime ideal inthe Dedekind domain A and therefore maximal
isAppfieldaThe integral

domainApprector space

with

an

The multiplication

element XE

Gp is an injective

i e surjective

endomorphism

every element inOpFoOp

Op is a fin dim

is a field

and therefore

has aninverse

P is maximal A

Propositiont

let p be a prime ideal
Then poOof A
Proofinsince

the factorization into prime ideals

A is unique there

Then HApoeTherefore

canwewithexistate

plp

ploie

find be p andseebe slportwithWe haveself

since otherwisesHAsp

por

If we would have poOP A Also
we wouldget

SPO ETOSOBut thenS

With XE On KT X

A

since A is integrally closed This would givep hTherefore po

beprime ideal in ASECHSLet pIn 6oA

have the unique factorization

wePipowhe Piareand herPiA

par wie different prime ideals in O1

The ideals PiPin Aa0p

areexactly those for whichPiliesorerfand write

We sayYpTeaeftp Defjtin

is called theinramification index of Pi over p

The1

degree of the field extension

fiscalled thePioverOp Ap

inertiadep

Theorem 10.4

Fundamental identity

Itcharktor
andIfofInkan

CL KPC A prime

poE lPieifiPinthen