53. GALIOS THEORY C eFL, *ie
ser all A cn) ie I cam be cons5d eznd a
Nogoa1 Pntesoom ethe o
PolgnomsalLs sh E C,
A colocieM
fibe entens®n k eF a hinshe nosaa field ovea E.
K s a spls g
n n s5em oP 4 k i a sepanab splstig fuld ble ovea E omd FEE-K,
ALso, s inu Sepana
ovA F.
3 thi s e ovea E.
s SepaaAbl E
noR al enens en o
finse
.
k s a
a n ambnmokplhiiak
elumor
o CKE
eu
Tom No
enrensev oF, ar u navg E fined.
also fined
etK be a i t e oRna lsaves
F
F E Ek < F . eloecr ohGnK | E)
E be an enrens5on o F, wh ve
And
Suoheld o E
,
b GCKIF).
Than K i a inse nozma\ enrenson G(KF).
a suognap
GaK/e)
** Sam ose GCk/E) »
Consssh o al Ahose auomorphims ta leavei dongs *
ned. MoRLova tuwo auo moephsms a n d T s G(klE).
eG
GCRF) b
CakF) Svndmu e ae
îsomozphis , E o
Le ad T elongs w the Sma
le cose o
Not
a Subsield oF tha na in me sama.
u
Cosee
GKlE) s G(rle)
( k IF) (K]F). HCnk[E)
Svina ka finste noKal edenSm F , K is a
splittie hield o same ser Lel aeE tl«) H) (x)
pelgnomials ) CH IS
an auhmphu
heb, K be a o k LAv e fined)
, Comveas el cet o all gwbgaoups o G (kF). e followrg propiy
mbmoRpms bn hold foa A:
aCk]p), wsch indmuL tne sanme isomoapoi on £
. ACE)G (k/e)
Kate)
) Tld) ae E 2. E KatKIe)
tola) de E 3. Fo A CK) H
H< G Ck|F) ,
[K:F)(Gn(KIF): (E)), h
T e (kIE) Ck:EE: acEand n Ca (kIF).
) md T belongs hh the 6ame l s tose b Gk}E\ uober co sek oaCE)
a noam4l enens50n
o F S a(E) i a noa
in Gk P) .E is a noA
Swoup o |F) . Klhan aCE)
than
SugroP G x l F ) ,
Remaak (kiF)/G(x 1E).
her s anA -to ene coRRODPNdence bln leyk cos es ol
Ge F) a iovar
subAOps o G k
1F)
The dsugeon o F.
GCkIE) io G(KlF) ad isooaphism owo a .
Yea e d Sa.
fields K v
olsngRam o
Subfeld ok Aving F fhned .
Dansom Pa-oo
Acld deininon ba
K L hnshe nokal envens5om o a i t is 1us the
GKF) na Galois gonp K ovF E s[wI) )
reklc)
Teos (Gelos tao
LeVe a hinst noaal emtenso oh a eld F We have to Þsove Ehar,E
whu
fild E,
aJ alois gaoup ak. IF). Fo
a
)
a(E) be h e sugaoup o a(KlF)
Efimed
F E K, l e a v ig
auoneah
a G
bavng E ned. n -
e ser oal cuth nrumed hae d dd3 E on Hhe clea i(KIE
ser all A cn) ie I cam be cons5d eznd a
Nogoa1 Pntesoom ethe o
PolgnomsalLs sh E C,
A colocieM
fibe entens®n k eF a hinshe nosaa field ovea E.
K s a spls g
n n s5em oP 4 k i a sepanab splstig fuld ble ovea E omd FEE-K,
ALso, s inu Sepana
ovA F.
3 thi s e ovea E.
s SepaaAbl E
noR al enens en o
finse
.
k s a
a n ambnmokplhiiak
elumor
o CKE
eu
Tom No
enrensev oF, ar u navg E fined.
also fined
etK be a i t e oRna lsaves
F
F E Ek < F . eloecr ohGnK | E)
E be an enrens5on o F, wh ve
And
Suoheld o E
,
b GCKIF).
Than K i a inse nozma\ enrenson G(KF).
a suognap
GaK/e)
** Sam ose GCk/E) »
Consssh o al Ahose auomorphims ta leavei dongs *
ned. MoRLova tuwo auo moephsms a n d T s G(klE).
eG
GCRF) b
CakF) Svndmu e ae
îsomozphis , E o
Le ad T elongs w the Sma
le cose o
Not
a Subsield oF tha na in me sama.
u
Cosee
GKlE) s G(rle)
( k IF) (K]F). HCnk[E)
Svina ka finste noKal edenSm F , K is a
splittie hield o same ser Lel aeE tl«) H) (x)
pelgnomials ) CH IS
an auhmphu
heb, K be a o k LAv e fined)
, Comveas el cet o all gwbgaoups o G (kF). e followrg propiy
mbmoRpms bn hold foa A:
aCk]p), wsch indmuL tne sanme isomoapoi on £
. ACE)G (k/e)
Kate)
) Tld) ae E 2. E KatKIe)
tola) de E 3. Fo A CK) H
H< G Ck|F) ,
[K:F)(Gn(KIF): (E)), h
T e (kIE) Ck:EE: acEand n Ca (kIF).
) md T belongs hh the 6ame l s tose b Gk}E\ uober co sek oaCE)
a noam4l enens50n
o F S a(E) i a noa
in Gk P) .E is a noA
Swoup o |F) . Klhan aCE)
than
SugroP G x l F ) ,
Remaak (kiF)/G(x 1E).
her s anA -to ene coRRODPNdence bln leyk cos es ol
Ge F) a iovar
subAOps o G k
1F)
The dsugeon o F.
GCkIE) io G(KlF) ad isooaphism owo a .
Yea e d Sa.
fields K v
olsngRam o
Subfeld ok Aving F fhned .
Dansom Pa-oo
Acld deininon ba
K L hnshe nokal envens5om o a i t is 1us the
GKF) na Galois gonp K ovF E s[wI) )
reklc)
Teos (Gelos tao
LeVe a hinst noaal emtenso oh a eld F We have to Þsove Ehar,E
whu
fild E,
aJ alois gaoup ak. IF). Fo
a
)
a(E) be h e sugaoup o a(KlF)
Efimed
F E K, l e a v ig
auoneah
a G
bavng E ned. n -
e ser oal cuth nrumed hae d dd3 E on Hhe clea i(KIE
