in Galois Theory What's ? Next finite extension well - - PowerPoint PPT Presentation

What's Next

in Galois Theory

?

We

understand

finite

extension

super

well

what

about

infinite degree

extensions ?

→

"

purely transcendental

extensions

hare

"

no algebra

"

→

"

purely algebraic

" extensions

have

LOTS atalgebra

Good

news

:

There

is

"Galois Theory

" for

infinite

algebraic

extensions

Bad

news

it's

more

complicated

EIK

( lip :p is prime))

is intuit

degree over Q

so

Gall Kla )

is

big !

Do

extensions

• f

Q

• f

degree

2

correspond to

subgroups

• f

Gal ( Kla)

that

are

index

2

Nis index

2 in Gall Kla) ⇒ NO Gall HQ)

and

Gul (14¥ ' Iz . But

such

N

are

counted

by

surjeotieexGnlll4@7-n7Lz.B-t

There

are

vncountubly

many

such

surjection

Are the

uncountable

many

degree

2 extensions of Q?

No

countably

may

!

So : too

may

index 2

subgroups

and

not

enough degree

2

extensions

• f

Q

Resolution

topolegize

Gal ( Kia)

and

then

focus

• n

subgroups that

are

"

topologically

nice

"

If

KIF

is

infinite

degree

Gall HF)

is

not

• n

"

topological

"

, but

it's

also

a

"

pntinik

"

group ,

ie ,

an

" inverse limit

"

• f finite groups

• EI

Kp # Kp. ←Kpk

• -
• a-finite

f

T

• i,

"

Fortuny

{ ioaipi

aiEkp )

the

inverse

limit

• f

Galois groups of

finite

extensions of

F

There

are

two

big

questions

• ① (Absolute

Galois groups ) let

F' set

be

the

separable

closure

"

• f

F ( smallest

separable

extension et F that's

algebraically

closed)

Gp

• Gal ( FTEPIF)

Infinite

Galois They

says

you

understand

GF

completely, then you

understand I

completely

Sad

news :

GF

is really really

scary

For

example

we don't understand

Ga

very

well .

Some

related

questions

:

⑨

how

are

groups of form

GE

special

"

away

the

larger

class of

pnetrnik groups ?

⑥

what

is

Gia ?

(see

Grothendieck)

④

for a

given

group

Q

,

does

GE → Q ( appropriately topologically)

By

Galois Thang

to

: does

F

have

an

extension

KIF

w/

Gal (KIFKQ ,

② gives

rise

to

"Inurn

Galois

Problem

BigQu#2 If

F

is

a given

field

and

G

is

a given

group

, can

we

find

KIF

so

Gall KIF ) t

en ?

I

let

F

be

a field

with

IFI -o

Then

" G

is

realizable

• ver

F

"

iff

Ge In

for

some

n .

What

about it

F

• Q ?

not

understood

completely

, but

→ any

Sn

is

realizable

• ver

Q

→

every

solvable

group

is

realizable

• ver

• -

For

a

given

F

and group

G,

how

can

we

go

searching

for

an

extension

k with

Gal ( KIF )' G ?

idea :

induction

Suppose

G

has

some

HOG

with % a Q

By

Galois Thy , if

we

had

some

KIF with

Gal ( KH a G ,

then

we'd

have

FELEK with

Gul ( Kk) IN

and

Gal ( TF) a- Od

±:i÷:÷÷:*::÷:÷¥:

Q

• extension

we

found

^

③ make

sure this

and N and

Q

For

instance

, if

FELEK with

Gallen)=K

K

and

Gal ( YF ) ' Ezio Ka

, then

Ka

l

L

God (

"le)

will

be

• ne
• f

Zettel

Ky @Kz

• r

Zz ④ Be

Re

F

• r

Qg

• r

174

This -

strategy

is

related

to

" Galois

embedding

problems

"

Then

are

connected

to

some

• f my

research

Basic

item :

suppose

Gul LKIFKQ

and

up Ek

Kummer Theory

it

:":i÷.. I :÷÷: :¥i

:*!)

(Fyi

• Ciri

. if

In

my

research , I study

a

relative

vision of

Kummer

Theory

:

viii.iii. I

I :¥÷÷÷÷:÷⇒3

• I

{IE

ri

:

a' era)

Resounding Theme

These

" Galois

modules

"

are

fur

more

"

Than

a

random module would

be

⇒

Gf

is

much

differ from

random

prelim't

group

G-

Lauren

Heller

and I

were

able to

compute

structure

when

churl F) =p

and

Gul ( KIF ) '

Kp ④ Kp