::::: E F EXT " Some - c x . PI roots of unity " - - PowerPoint PPT Presentation

Applications

II :

Kummer Theory

Lastttme

Proof of

Galois

'

Great

Theorem

Necessary feel

Thu ( Kummer Theory)

÷÷÷:÷÷:÷÷:÷:÷÷:÷÷÷÷

Some

x

"

• c

E F EXT

PI

we

already proved

"

⇐

"

in

the

roots of unity

section .

Today

we

do

" ⇒

"

Setup

:

Gul ( Khs) as In

and

wa EF .

So there

exists

re GallKIF)

with

Lo >

= Gal (KIF)

and

Irl

• m

with mln

(ie ,

{ id.hr}

• -irm

" )

are

district

elements

in Gakkai)

that comprise

Gal ( KIF) )

Strategy

:

① F Pek

so

that

rfpl

• wmp

where

Wm is

the

primitive

mth not

at

unity

wi

'm

② show

xm

• pm

c- FIX

)

,

and therefore

×

"

• p

"

=

x

"

• (pm)"m

c-FIX)

⑦ Argue

that

k is the

splitting

field

• f XII

by showing

K

• e FCB)

treats are

{ P , wnp , wip .

• '

p )

Step

since

Eid, r , o'

,

" )

are

pairwise

distinct, linear

independence

• f

characters

says They

are independent

• ver

K

no

expression

• K. id t k

, r t - -t km, rm

• ' E O

without

Ko

• k,
• --
• km
• O .

On the

• ther

hand ,

since

rm

• id

,

we

have

TM

• id

E O

We

want

some

PEO

so

That

rlp)

• wmp

If this

failed

, thus

(r

• wmid)lp)=O
• nly

when

13=0

But

• bserve

rm

• id

equals

( r - id ) (r

• wmid ) (r
• Wai id)

• ( r - wmm
• ' id)

( this

comes them the foot

xm

• I :( x
• Nx
• wmllx
• wm

')

• - -Humm)

)

since H

• id ) ( p)
• o

for

all

p

and th

fact

that

fr

• wmid ) Ip)
• O
• nly fer

13=0 , we have

(r - id) Cr

• wniid )

• ( r
• wmm
• ' id)

is

the

zero

function .

But

expanding this poly gives

a dependence array {id, r

,

. rn

")

→a-

Hence

there

is

some

1340

so

that Hp)

• Wm P.

Step②

we

want

XM - pm

C- FIX] . This

just

means we

want

pm EF

We

can

do this

by shoving

c- ( pm)

• pm for

all It GallKIF)

Since

Crs

• Gall KIF)

, this

is

equivalent

to

r ( pm) - pm

• bserve
• 1pm)
• Hpk
• Lumpy
• wi pm
• pm

steps

we

want

K

• Ftp)

Note

:

since

rlp) : Wm PFP ,

we

have

p # F

Since KIF is

Galois

,

we

have

[K :F)

= ( Gall KIF) I

= Kr > I

= m

We'll

show that

9413,132 ,

" )

are

an

F

• indep

collection

Hence

[ Ffp)

: F] > m

, and

since

Flp) Ek

and

Lk : F)

=

• n

, this

forces

K

• Ffp) .

Suppose

instead

That

{ 413,13

'

,

") is

dependent

• ver F

WLOG ,

assume

we

have

a

liner

dependence

Colt c. Pt cap

' t

• t cm . . pm

" =D

all cief

That

has

the

minimal number of

nonzero

c.effs .

Assume Tmt

fer

c.,c, ,

above , Tmt

r is

chosen

to

be

the

snarliest

index

with

• r to :

r

rt'

Cr Pt cap

t

• t cm . . pm

" =D

all cief

Dividing

by

P

"

gives

①

Cr t cap t

• t cm . . pm
• " r
• O

all cief

Apply

r to both

sides :

rlcrltokr.ir/plt--trlcmn)r/plm-r--rCo)

{

u

②

Cr t Crt , WmB t

• t cm, wmm

" -rpm

• t - r =p

If

we

consider

① - ② ,

we

get

back

a

liner

relation

at

{ 1. P .

" ) that has

fewer

nonzero terms

→a-

pg