II : bustling :i Galois theory ) ( Fundamental Theorem of Thur - - - PowerPoint PPT Presentation

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Fundamental Theorem worked examples II : bustling :i Galois theory ) ( Fundamental Theorem of Thur - ids.ba , The and f satisfy A. F F functions - ::i:: Galois ift Ciii ) KIF Alk ) of is iff FIH )

SLIDE 1

Fundamental

Theorem

worked

examples

SLIDE 2

bustling

Thur ( Fundamental Theorem of Galois theory) The functions F and f satisfy

A. F
ids.ba,

÷÷÷÷÷÷:÷:÷i÷:÷:÷÷÷÷÷

:i÷÷÷

Ciii) KIF is Galois ift Alk) of Civ) HOG iff FIH) Ip is Galois .

SLIDE 3

Y

Use

the

fundamental

theorem

to explore

Sean

familiar

examples

SLIDE 4

Consider

EHR

. We've computed

Gayle

= Eide , complex conjugation : That bi) = a

i÷÷÷÷÷÷÷÷

SLIDE 5

let E be splitting field for x3

HI .

we've seen Gal ( E/④) a- Sz , with rt GulfEl ) acting as an element

symmetric

group an

{ Vz, wVz,

w't } L , L2 % lid) E i:i÷÷ : S3

µ.ge

are ④ ✓

nly intermediate

Sub (G) extent""! Lat LEI )

SLIDE 6 What is 112)

c Gall Ela) ?

Should be the isomorphism that sends a, t > az Lz ( → a £31463 So : what is F- (cans) ?

Okla)

: fir)

Similarly :

Flash) ?

Okla)

wut is 9- Kay 's) ?

14g?÷÷¥÷÷:

⇒ out

SLIDE 7 Exe suppose Gall KIF) ' En . Recall : subgroups of kn are precisely cyclic subgroups

size d , where d is a

positive

divisor of n . Moreover ,

H . . Ha E kn , then we

get

A. a- Hz

iff

tHill that . By Galois they , we have Lat ( KIF) is "dual " te this structure .

SLIDE 8 So

Gul ( KH ) E 742 (divisors are : 12.3.46, 12) corespewdj subgroups H ,

{ e) , Hu
G

. . .

let

:÷÷÷÷÷i÷

Sub (G) Lat ( KIF )

SLIDE 9 One

special

bservation

Cod

( Galois

extensions have finite lattices)

KIF

is Galois , then

that CKIFH so

Elements

Lat ( KIF)

are in

bijection

with subgroups

at Gallerie)

, and we have at most

214144Mt

. . 28k '-F) subgroups . DM