Ex TENS1ONNS
31 nbqs ALGBRALC 1 E finshe e n e n s n fidd a
a
field F, ard k
ield a E, then k i a finie
algebant exensbm is finsle entenon
n exten son sield E a fidd F u an
enenson o , m
ovex F.
elmant E 6 alychaue
eenn eld E q F f i o e dimunsAM
n a
F (k:E][E FJ
Snsteenlens m dagkL2
spi E, ttun E is a
Vec oves
a a s for E as
t l , 2 .. . , n b e
ove F. he
I
E vF ove Fadlat tha sk pi j : \ ,
k shall Let me degKea no vec spa
LE Fbe k as a Veto Spaua2 oVu E.
basis foa
be a
forma a s
TeFI FF. ws shoeo fhal melams n ; Bi
Vi ewed a
Vetwa spaa ove F
ueL )
field f is atgeban
b K.
an
A finie ecenson ield E q a
elamun
er be amy
have
& basis foa k ves E
Sme B, foarn
xe
basis to E ee
oe go3m a
LetE.F n tn
l, , d . . | a conno Swwo 'S
i l ndependsr deenk.
=o nor all
t a, + t . a, a's 0,
" u
a non 3u0 polupnon
a (1 bt , u -. + h " - o
iP)
SpAns
Kene F
, E Aamains us to shous ha wn elmenk a
Phoof
enn ns>n ield oF omd eE e algebrae
indaperddy bve Aet E be m
oAF. 1f dag Ca F) nhan C«) ts nn-dbmunsan
Suppose at,
Pj) C F. vecoa Spate evea Fwob bss
CA F
Fualu mo evu elenmen pe f la) is algebraie oves
)F rd
dagsa. ( p , E) dg l«IE)
F ) >n, t n dag aE)[F):F}*n
her
i a rdapendero
Cij 0 C s lunaay indaperdaor) B e FC) > FL) FC«)
F FCF) F(x)
hus hun b aave miorem ( x )
e fo a aris fos k bve F.
wwy k is a fiml tnrensiom fild F wd
FC):F (Ft):FCP) (rce): P
mn
dodlas F))
daa tp, F)
Condlo
f is
a field for i-l-. ,7nd . fiod àlgaea od bass Sos ma, sollotoings
enkens>
Fi, then F s a Sinse tahens 1on .
md v a , 2) ovu
F-[ -{). 2 -2
2 is he peluynoial
CoAoloa
dagAaa 2
E an enenssoo iud , FaeE algebe
oveA F, ad BFC), hen dg dan 2, a)
CPF) dovides dag P)
31 nbqs ALGBRALC 1 E finshe e n e n s n fidd a
a
field F, ard k
ield a E, then k i a finie
algebant exensbm is finsle entenon
n exten son sield E a fidd F u an
enenson o , m
ovex F.
elmant E 6 alychaue
eenn eld E q F f i o e dimunsAM
n a
F (k:E][E FJ
Snsteenlens m dagkL2
spi E, ttun E is a
Vec oves
a a s for E as
t l , 2 .. . , n b e
ove F. he
I
E vF ove Fadlat tha sk pi j : \ ,
k shall Let me degKea no vec spa
LE Fbe k as a Veto Spaua2 oVu E.
basis foa
be a
forma a s
TeFI FF. ws shoeo fhal melams n ; Bi
Vi ewed a
Vetwa spaa ove F
ueL )
field f is atgeban
b K.
an
A finie ecenson ield E q a
elamun
er be amy
have
& basis foa k ves E
Sme B, foarn
xe
basis to E ee
oe go3m a
LetE.F n tn
l, , d . . | a conno Swwo 'S
i l ndependsr deenk.
=o nor all
t a, + t . a, a's 0,
" u
a non 3u0 polupnon
a (1 bt , u -. + h " - o
iP)
SpAns
Kene F
, E Aamains us to shous ha wn elmenk a
Phoof
enn ns>n ield oF omd eE e algebrae
indaperddy bve Aet E be m
oAF. 1f dag Ca F) nhan C«) ts nn-dbmunsan
Suppose at,
Pj) C F. vecoa Spate evea Fwob bss
CA F
Fualu mo evu elenmen pe f la) is algebraie oves
)F rd
dagsa. ( p , E) dg l«IE)
F ) >n, t n dag aE)[F):F}*n
her
i a rdapendero
Cij 0 C s lunaay indaperdaor) B e FC) > FL) FC«)
F FCF) F(x)
hus hun b aave miorem ( x )
e fo a aris fos k bve F.
wwy k is a fiml tnrensiom fild F wd
FC):F (Ft):FCP) (rce): P
mn
dodlas F))
daa tp, F)
Condlo
f is
a field for i-l-. ,7nd . fiod àlgaea od bass Sos ma, sollotoings
enkens>
Fi, then F s a Sinse tahens 1on .
md v a , 2) ovu
F-[ -{). 2 -2
2 is he peluynoial
CoAoloa
dagAaa 2
E an enenssoo iud , FaeE algebe
oveA F, ad BFC), hen dg dan 2, a)
CPF) dovides dag P)
