II : " , an ) for algebraic lemme If - Fla , E - element - - PowerPoint PPT Presentation

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II : " , an ) for algebraic lemme If - Fla , E - element - - PowerPoint PPT Presentation

Mar 23, 2023 149 likes •279 views

The Galois Group worked example A II : " , an ) for algebraic lemme If - Fla , E - element regal CEH ) then elements an 4 , . . , an . , . action its a , . . , an determined on by . is . . . , Olan ) , knew the values Ha

slide-1
SLIDE 1

The

Galois

Group

II

:

A

worked

example

slide-2
SLIDE 2

lemme If

E

  • Fla ,
", an) for algebraic

elements

4 ,

. . . , an . ,

then

an

element regal CEH)

is

determined

by

its

action

  • n

a ,

. . ., an .

Ie ,

if

we

knew the

values

Ha ,) ,

. . . , Olan ),

Then

we

know how

t

acts

  • n

any

EEE

.

key

An

F

  • basis

for

E

is given

by

③ { me

. . . - men : oeeiadlirf-ca.ai.im))
slide-3
SLIDE 3
slide-4
SLIDE 4

EI let

E

be the

splitting field fer

x'

  • LEAD
.

last time

we

recalled

roots

are

L,

= 352

,

Lz

= w UI

,

23

= WZ 352

where

w =

  • HEI EE

and

w'

  • -t=w

On

homework 6

, we

saw

E -

K .

, az )

slide-5
SLIDE 5

Big

question

:

what

is

Gal LE la) ?

Two

answers

:

①

group

Theoretic description

②

more explicitly lied

to

description

  • f

elements

Tt Gal ( E IQ)

as Anaheim

slide-6
SLIDE 6

whiten

Since x3

  • 2

has

3

roots

,

Rachel

told

us

that

Gal LE la)

E 53

Additionally

from

homework

6

you

know

[ E

:

④3=6

.

From

last him

,

since

E

is

a splitting

field

we get

lGallEla H

  • LE
'
  • QT

F)

= 6 .

⇒LGullElQ)=#

slide-7
SLIDE 7

Can

we understand

how

re Gul (ElQ) act as functions

?

  • iii.iii.

" ""

f¥¥÷:÷÷÷÷

::

  • . -

Also

irreaka)

= x' tax ta?

Q

→ K - basis to

E

is It, q)

slide-8
SLIDE 8

Can

we understand

how

re Gul (ElQ) act as functions

?

  • E
  • la , da)

Rebeccah

showed

as

that if

we

want to

kilo,

"""

I :*:

i

:O

:

.

  • basis for

E is

l

T

: E → E

,

we

can

Q ftp.2.qa.a.az

"

climbing

up to

E

via

K

"

slide-9
SLIDE 9

Qi -basis for

E is tha , airman , dial

we

can first

extend

idea

up to k by sending

.

d.

to

any

root of idairro.cat/--x3-2

4

,

1-72 ,

  • r

2, t L,

  • r

4743

4¥ We

can then

extend

4 by sending

La to any

not

  • f Ylirrk la)

)

  • x' ta
? xta?
  • O-

For example , if

4k .)

'
  • he , [

Roots at

this

are

a

.

and as

, so

then flirt,aka))

  • x'thxtht

either

dats d

,
  • r 221-223
slide-10
SLIDE 10

For re Gal (Etta )

determined

by

rca ) =L

and

  • (a) = Lz

,

how does r act

  • n

a

genial

EEE ?

A given

EEE

has

a

unique

expression

in th

term

e

  • q , I tqratqga-tqyhatqsa.az

+ qua? K

.

S .

what

is

r ( e ) ?

slide-11
SLIDE 11

So

e

  • g. Itza , tqga-tqyxatq.a.az

4) of

+ gain)

= q , tqzrkiltqgok.Ytqyokdtqs.tk . )rKr)

+ 26 ok . )

' rca)

= qtqztztqztzhtqydzt Esau, +962543 .

Note : this

is

no

longer in terms of

"

  • ur
"

basis

slide-12
SLIDE 12

Onelustbkuhin

,

we

viewed

at Gul CEIHQ)

by

using the

intermediate

field

K

  • Okla ) .

One

could

use

any intermediate field

wuthin

El

.

Upcoming

problem for

homework :

can fellow

then

ideas

by

working

"

through

"

L

  • ( w)
.