(1-9) INTE GRAL ExTEN SIO NS

. DesinitHon 1.9.1’ Let be a sing Ond fX) e Atx]. hen

fx) said do be
be a
a monic
monic palynomie
the caasticiont of 4he lhacUng term of fx) is 1
" DefinitHen 19.·2 ’ Let A lba a ing extension. An
elomant Baid to be indagral
$x) =0/ ohene f0 eRCX] ia monic olyhomialie
a x + an =0, lohet aie A, 70

Poroposiien 1:9-3 ? ’ Lat be a sivg extensieh. They

the fotlouaing equivalent.
1 Over AR
Ainita genanatod RA- medule
3 RACX is containad h a Aub ung C such that c is aaf&
A-modale.

Proof :: ’ (1)>(2) : f xea

eB is indenal Ove A then
fCX) =o AC] amonic
monic polgnomi«Q
+ a,ya+ -+ay
(14i4n)
I,, -- - À- hed.lo

(2) ’ (3) ¢Cx] c C Exerie

Take c= 4C*I
R
(3) ’ 1) Let Cs, --- Cy gentatesC as À- medla- Let x; -}A;
----+ yCy i=l
2C2 = 21 G + C t + Ar
Cy i-2
i=3
 Hanca xx
xi, efpoweM All
=0 x2) f(x1, A
indegra! in lie
A
Oen
2n= for
provali Inducte by
frous, We Paoof
n=1 Foy
meclulo. tad
A- gonena fntay a is-,Kn] ACKi,--
then bwsnAindagal
neß -
Ki-, entansten
4 a a be Lat
B
S 08/10/11
extensien phen J=In A
polynomial
and monic Condal
a nin idoalthe
an be
ALYD
A[] Ic and ing a be Lot ’ 19-4 Example "
A:OVen datominant the
0 dotd
= 1eC, Sinca
o =0
=0 (d)C det
C
-X23 I-22
R-A-A12
(x-A)Cr
=0
Gr
-r
=0 isCs - C2 Ai2 (-1)
- C1
 (c)
"ProosiHen 1:9.6 Let B be extension The At of
all elemandA of shich aa
R

Psroof: Lot C ba yhe set of al eloments ef which ane

lot e C "Than KA Cx,y] afnitay gant-t)
RACXty] cRACX,Y) 4 2ACxy] cA C*,] .
intagal qusyA and Aence

"Definiien 1 g7 ’ Subing cC dafned in

closue of Ak tn e
Let 4he
th notaten be as in g.6 |·37:
han KA Bi Said th be an
irdegal We that

A =C hen A is said to be Irdaaaly oloAd in B.

R
3. RA =C ba dealn B 4ha
he quotlent fatd of A, Hhoy
QA is sald to be indagrally clokad.
.Examkos The
inteal in
he in Q(i) 2iz
4he
xing algehoaite
Poobositien (-9:8 :’ Ond B c
thon

EC then Bahshel equaton

 over A'=Acai, a2, - --,an], So
so that
RA'CXis Sinca
medula
Slnca

ropositien 19.9 ’ Lot AGe ba an integra! extanaien let

I be an idual of B and J = InA
indegnal extanion. S Exer
boof:’ Let e B/r oheas ban donate a0duction nmodlulo J.
Since Over A, x Satafes a monic folynomia
ie + an=0 Lohere aieA
Ghoing modulo I W e
9-2
+ a, =ö
OVe

be an
b a tmulthlicahu clsaad Subsat of A
an tegal cxdonala,.
Let sB
lot XeB an ovn A Than
Fhen

divide bath aidos by s

Phobositien 1 9 11 :> Let ACB be an

domalnd Then As a feld Jf and
Rald.
 dealfB ofA maimal
a ba þ
Coxollag
Tat AB
J9-12’
Laut
t
Qn
be ful| d a is A
AHonca
o,¿ +
=1 =0
sidos both
aeA shtne +a,
Phan
nt. re hon-
hor abe eA and feld aa be B lat ConvenAy
fald
( La,
=0 an-)+
an +
thealemant hancs
hanca Qnd eA a fcld a s A As
that follows it donan ais Sinca
B
eA, a;lohene =0 an +
f(x)=o Indegtal
A
OVen
-7070 non 4
and feld a is SupbeAd
A Paoof:
A