Theorem 5.6.1

(1)

UR-V2

FiEDS FiwiTE FiEios

heoom 6

AP ^k ^a

deld

^and

f o.

distnc Qutomophfema o k thon t mpossle

an not al Zeo n

l«

to fhd elmens a

Ato Cu) +aa Cu) 4 anOn Cu) - = D uek

Proof

Surpose e Coulel Fend a Set, elements

an

.Inot alu 5 hen

Then _we

Could tad Such

^a

relakon haina

reLw non-zeo tenms a»

Possible On emerobain

Coe ^Can ^{a u m e} that

ha mfnimal elaton u, aoCu) +

am _m Cu) ^o

wher

_a _m

to t h e n

aCCu) =o fr

m CLere

Caual

au

ue k teading

^to

a 0 Contray

^to

ayumptom hoe

ö an

eloment

Thus ue a axume thatF

Cek

⁹

olc) *m c e).

Stnce

Cuek

fau uel< Telahon O

^must

aluo

hold for Cu,hat a tu) t am TmCu)eo

fon auuee Usrqthe hypothasis hva he

(2)

Os ae Qutomoophtsms ot k, hLvelaton

eomos

a CE) o Cu)+a20aCu) o Cch +

amo m Cu)

^Troc)

Mulktdynq delatonby ojc

o C e ) s Cu) 4 ao tu) o c c ) 4 - .am Omtu) c e ) o .

ard Subtaalttng

^he ^olakon ^kom

eld

a2 _o

lc)o

Cu) ^- ^aoblu)i

C) + .am Om COlu))

as t u ) lac) o,C) Qm Onl u)lomCe)-,cE3))

we

put bi= a? C orceS}- co) fo fo a-m

Ahen

_tho

b

_{a e}

n

_k

bn

am

mcc)- Ce)) +o SThce

amo

m Cc - Ce)

Oo Cu) om Om

Cu) a o for all ue,K

Th&& Contra

^to ^oun

CUSuMPtton Hence paed

Deptton

ë a gooup of aukormo phüm qH

ton the e Aatd oG the Set qall

elements ae l oCa) a

r all

^ae

G

Lomma

The oces reld e G a Sabfield

(3)

ee PocoLet

^{c t} ^a,b

be

ⁿ ^the

txed ftelJ eg teld G

Thus fo all ae ,

hs

, C a ) a ?Ch) »b. Bu

thon Cat b)

⁼

oCa) oCb) a atb ard otab oCa)otb)

⁼ ^ab

ence a b

Q ab a e

i n

Pthe fored ted G,1f bto hen otbh oCb)b, tence bolso falls

ⁿ

The fred teld o G Thus

^we

hae Veifted Rat

the

ed fetd o G&ndaed a Subleld o Delnitton

e t ^k be ^a

teld ard F. be

^a

Subpeld

o Thon the he grcup f automophism

elativo writen G CKF

^he

set q al automovphis m

^K

luna e e elomont

^ü

Ere).

that

8 he

auomophiäm

^o

q

^k

s

ⁱⁿ

Gt) Tt Ca) =

^d

os eey

^{e F}

kemoa

GCe,F) a 8ugreup the the grou

all autombaphum O

Romark

k

Contains

he eld o atcevals umbors STnoe k

^&

Chazactoriste

^O ^ard

iE

Fo

e o o See that the lre Fld of

(4)

nt aur autoophism k bang

^a

Teld, must Contatns fo Htence owy.Patioa umbex & lelt Ptxed by e very Quterahism

talt

EXample 661

the teld tomplex numbe

let ^{b o}

nd l e F be the Pted ea numbes Cue Coreto GcK,E)

anu auleoophusm ,

Since

Hencerc?)» t i .

i n addi tton, l e a a eug e l hum koa

r e d h e n for anu a b i , Whore a,b R humket

Catbi) = t a ) o t b ) s t i ) = atb?

Each o hese 9sblitkes Nanmely he Mat C a tbi)

⁼ a4 bi 3

d Cas bi)

^=a^-b?

delnes

an auteroophim R

+being

^an

dentit automorphisro

Comple Confgafkan.

hus G Ck F)

^a

ooup q brde hus 2

b i in Faoe teld q 4 ClkiF) thon

aAbi=

^ola

tbil = a-b

(5)

twhee b=o = aHbî Ef.

Ths dhe e d eld o GCKF) Poei spln

tsel

Excarmple 5-b 2

let Fo be tho eld Qf

^Yakonal

numbeu

ard let

^k

Fo(V5) wfero 3

^s ^ho ^0al

ube

Evezu Evey element n k ü q he foom

d + 3 tdo.3)

^whoie

ddod o

^aue

uhon humbers i o an utomophöm" ot hen

o(3) olav)*) = ot) -2. hence ( )

Mus also e a cubie hoot 2 lying înk, hououex hero ü onlg ore veal Cuhe Toot 2 ,

houwouex

and Shce

^k

a Subfteld r e eal teld,

Cwe nust haue rat

L3Va) -31à

^But

hen

1

Cdo +i 35 4 d (3U)zdo

⁺

3 6 +daa

.

hat & the fdclthy automosphis m o

^k

we thus S e

that CkFo) Consists 6ny o ke Tclont op and H case the red old

GC fo), notFo but n fact lage

o

beirg all of

(6)

Fxcmple 5-63 Fc lemma 5.62

feld of ahom htum be ocnd

the,

et Fo, be

ard to Sattsfies

nYs,

e t t te = e

thu

w=)

ineducible

h e Polyromial "+

Thuu

^k

Fo Cw) cA dearoe 4

^ove

Fo ard e y element în o tke fomn.

k FoCw)

⁼

do

wH

odo 0Hduo3

where do, Ancd dg

aee n

Pn Fo fo.

^Nou

tor Cen autommehism

w ) /1 Snce C1)

1

by Aved feetd del-J.

Aherefove, o(u w")

OC)

Cohence

_{l w )} u

also

_he

" roo uniy

n

Consequence

^s

TCw)

^Cah

only

^be

one. qd,

e t

^us

delene appin gs,

^o

by in aerally,

dot d t odO+ dauo)

= do t

di(w)

⁺

2 r 3, u

(7)

Eouch here2 deltnes

^{a n}

autormephism k

e

GCkFe)

R C o m e t d

úagidic qhoup

Phexelere, Since

pndex oleloamired by w), G(r, Fo)

aooup o ocdo c i h

:-i,-13 .

GCefo) a jélie group oda 4

Dhe Can

easily Proe

^that ^the

Ficed Aold q CKI Fo) d'Fo Ttselb

he Sukgoup A= o GCkFo ) has

as

7 ts feed ald he e alt elonenrs

o t d a l w) which

an etenian a

to o degvee2

Theovemn 5-64

Let K

^be ^{" a}

homal eatenion of

F and

^lot H be a

SeebgrouP GCk sF) ;

let KH= {nek /~ca) - a for pll eHG be

he hixed fteld o . Then

k ; kHJ - o CH)

2.

H G Ck,kA).

(8)

Jo Pautculax, uohon H GCk; F), /kiEJ

GCKF))

Proot

Sice eery SPce olemenE Pn H loaves kKy elemontsse teld, ceatatnly

^H ^e

G Ck, k)

By

Vheorom .6.6.2

we Row that,

Tk:M)

^o(G

Cr:H))

^z ^oCH) ^{w e}

hae the nequnlites k : KHJ>olG Ck: rn)

2 00)

c o e Could 3hoo that K H

COe

oCH} TE

tolloco thab otH)» ol GCk,k)) and

^a ^a

Subgoup of G CkkH) havng

^{c l o}

CO ould mmodfately

that of G CKrkH) twe would oblam that H

GCkKH)

⁸⁰ ^{w e} ^must

mesel

^shouo

hat

i e A ) = oCH) o Prove LeLy theng

Theorom 5 5 1Theee niss

a n a E K

Seuch t a t k k H Ca) hi's a mut Thaejo

Satisfy SatTsy

^an ^cducibe

Polynomial

^o

kH degroe

^m-

lk: k H,] and

^{n o}

nontiua)

olynomua loua degree 7h eovem 5:13 Jet

^he

elemens H

^be

o o o

Chere t h e the den ty GCkIF)

h OCH) Consiclen he

and hence

(9)

elomont aay Sy mmot ee Funchons

a = Ca),oo Ca) On Ca hamely

O ; Ca) 8 2 Ca) + On Ca) . Ca)

p

^pt

(a) o Ca)

dn = O Ca) O Ca) h la)

Each ? hvazfant

eundoz eLy oEH Ths biy tho olen KH

all elemens o kH Houcuer a

as uell o'b la) --

^{C a )} ^a ^odot ^he

he

as

PC)= C a )x-ola)).

o lynonmial

C h Ca) ) = . - 17 + o/a . +(-1) di

haung Coebclens In H By he natue of a thå force h 2 m - k : kH where OH)>|

L

Ie: k). Snce

^e

alaoady know that oH) k : k H ] we obtatn oCH) = lK: kH) he desired

^COn

cluston

when

H GCk F) by

^{h e}

nerali ty q

vex F kH F consauenty foyhus

Pateculas Cae lkFJ =olG tk, FD).

(10)

pheeröm 562

It ku o tnite o ntonion q f . thon GlkiF)

&a a inito q2oup ard fts O vdor o C GCkF))

Satses olGtk F))

^<

[k;F

Po ro

etK:F -n

Suprose h a t u ^un

&a

^ba's

qk

bue FSuposc

Can thd h+ disknc

On+ fn GCkF) automobphrsm ²

Byhe By

^Cord

lay

^to ³ ³ The System a

homogenus Lin aa eacaron

ⁿ ^{he h+)}

n

unknaoNs

+ O 4 CU ) Xn+ ^I=0

Ccu Xp

⁺ ^oQ

lu) rot Cut)Cx,) 4 lui) *a

Oun)X +ob Cun) 2t

OhttCun) Xnt! O haa non-trvcal Solutton ( o t all o)

X n4I

^an+| ^Yn

O

o

Cui)+a oh Cuf) ant 1nI Cu) O- (t

o r l , 2 . .

ⁿ

ele ment

^In ^F

lekt med by

Cach O ano 8hce ^{a n}

abitoay elementn

k U the fom t n Cn otth

n n F hen

Om the &ystem ea uakon ) _{Cwe geE}

a Ct) +.

_an+ ^{+ C 6 )}

=

^o ^fo

al

(11)

thd Cotrodi chs The e s ult a tek ^But

Theoremn

^{5 6}

5.6 has

been Poud

Theorem

DeynilionHeto q Ratkoral Funckon

ield, F l * n) be on Fitag rol

tearal

I P F á a

As Such wve

Corstauc ?6 freld

olomaPn

uokens

Cwe cal thu he teld cf atiora) funcH'o inX n Ove F and deroto P

by FCx, a2

Defenition Summe the'e

akona Functon

Jet Sn

^be

he dmneec Jroup e deqvee

h a n o F (xj *. Xn) Peld o akonad funeron Dehe a

mappng

^F ^C . n ) on to hsel

als

^I+

Constr

^all

Yatonal

HeencHons Co tn) Su ch hat 7 Ca Mn) =

Y c ) ) 2 oco) o-cn)

) tor

But hese are Precisoly those elemen ts xn whech ale Khown as he Symme+tc atonadunckbn

DetiniHon ield ot Symmeic aona JuncienJ

Been Berng he xe teld q Sn he

foom a Subfteld e F (x . . n ) Callee

(12)

foeld Symmetytr nalona tiuncon hith

e

hall

dentte

y Thoovom 669

Let

^be _teld _nd ^lot

(

^{n )}

he tho reld o alional u n t i o n s n n

i I e eld

ver Suppee

Sypometaic ^aionalJ unciion h o n

( r ( . Yn)s)n

Stymrnodsr oup

( C (Y 2n),s) S n

^ho

d oqnee h

3 , f a an O1 ho elemon tau Sqmmete

then FCa *..an) tnctons In

H FC .n)úthe Splitttng fteld

^oue

Fla an)

2

an

ho

Poly romialE at"aot

^.

4C-) a oof Sn

ce ho

ooup

^Sn ^o

o anoup

a

utonp

_Smy

F ( Tn) leouing3 fred, 3n eGCF (M)Xn),9)

Ihw by heorem 5.2

P a tenite

^exdension

q F

^dhen

GCF)

a f n i te qpoup ard T ovder oCG(K/F)) Sat Tsfies

OC GCk,F)) [k:¬J

T T FCX n ) a tnide e x t o r i a n o t S,

then

oGCECa an))) ] Ftx,. .. 7):9

O(Sn) C n):S)

FC n ) : 3 Could Shoo

^dhat

F ( . Sn

Ao w oe

(13)

Shoe. F Ca an) u a Subfteld of S, ue CouL

S fhce

haue n!

IF

^CXI**.^Xn)F^Cai ^an)]

w e w o u l d get ^{t t a t}

FCxj .. *i):$J=n

Thon

and

H e n e S I F C O .

an)

o r d So

S=F

^Ca1a,)

a

-hally,

Sn

^GCF

CX1

^*n))

all in S, heMeld FCa.Q

Ore

cuh Sth ce

^an

obained by

^adofhinq ^the elemontaxy

Symmatie

must lte in S funckon Qn ^{to f}

Hence S= F Ca an)

v

^{Thu PCt)} ^{t h e}

Polynomlal q desee

n

^a

FCa an) Splis ^O ^a ^Prco

uch Ene

i n e a z

facox

Ove

F C

^{n )}

Cannot s p l i t o v e a

Pooper Subleld FC

^{n )}^cwhcch C o n t a r n s FCa ^{n )}^foy G

Sub pteld coould than ^{a e}to conen

boh

^F

and ^each the

oo s

^o ^PLt) ^namely

n

T h u w e S e e that

FC n

^{h e}

Splitng

e l d o the Polynomial

PCE)= " - a t ^ . . ⁺ C-1)an ^{o v a x}

F (a

Hence he pooo

Detnition:

Let fex)

^be ^a Polynomêa fn

Flxj

^ard

e kF'

^SplHng

teeld

^{o v a} ^.

T h e

galofs

3ooup o fcn)

^he ^9oup

GC k

^F) ^all

he

automd p h t s m a

eaving evy elemen F dLF

ied Then th Salofs group 4t) can be

(14)

o t fC) deGCk)F) thon dla) u alysq Comideed a a group qPemukakens of its

root FCn

L - s ) - - s s ( z - )

A s z - )

o

L7(z)&o B»t 7

- b a anHsod

a44 o

5 D

pouop z