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Assignment in^ infinite
Galis extas &
(^1).^ Let^ HCG=^ 4(4/K)^ and^2 -> G (^).^5600 :^2 ***^ - (^) (2).
ast
Solution (^). Let -L (^). Then 2 20(xx)^ = x
- for (^) all +H.^ Hence^ xs = (^) - /Tc(x) (^).^ We^ show > (^) (x) + L. Let H (^). Then It^ EH- 58 en ( (x))^ =^ xx)
E) We o^ )- x =H
Thus - (^) 2x) -> (^). Conversalylet y^ -> 24.. Then
fteH
,
- a (0(y)) = w+^ (y) = c (^) (y). (^) SorsD=L
2.^ Let^2 =^ =^ G^ (2/k)^ and^ K^ <^ EsL^ bes^ a tower^ of
fields
. (^) Show (^) &(20(E))= 8 9 12/5)= solution. Essous to^ show^ that -14/020914) . L Let (^) H = (^) G(4/E). Then (UHC -)L^ (E))= e) = (^) 1914/01E)
3. Let^ H
,
H' ba^ closed
subgroups of^3 and [H (^).^ H'3^ = subgroup (^) generated by
Hand H'
show that (1) Yes!
H2s 1 H solution.^ It^ is^ alcar^ that^ L^ -1 2^. Let x^ =^ +^ and^ y
- (^) and (^) - -> Hist! Ther (^) r(xy) = 0(x)0(y) =
xy-
So (^) we have
the
equality . (2) By Galois Eresespradance -(4) 13) = G(4- +, 3)=^ +^13. Y and^ It' are Galo's^ extensions^ - Determine (^5) = 4)^ 4 2). Noter /Iis
a Galois extension . So 5 is a closed
subgroup (^) of -12/1)^. I 11 I I
,^3
! (^) Ne (^) need to sbow rest's Et (^) :^ I xen (^) them ~
- (^) [H , A']^.^ Hence^ <He ([
,'S
- (^) Get E , F (^) be interendiate subfacids of the Galis (^) extension (^) 2/K (^). Let (^) H , = 9(2/E) and Hg =^ 3(4/F)^
. Shar that
(1) 3(YEF)^ =^ G(y)E)^ +(v)=) (3372/15) =^ E, H-]
⑰
- (^) Show that^ an (^) subjers of
are-
(^3) 72F) whose^ I FSK is^ a Finsite extension in L. =^ Solution G .Let t^ bepai^ Theto his to (^) , thus a is digit wi o cosets is "Sida (^) Fixite une
of left^
costs
of
H . Stance [4 :^ H] Say.
Thro /
H = 4-
I finitely maxy left
cosets is
closed So^ - (4/() = E =^ H^ and
(^24) /I is a te (^) extension since A has firs finite
index in 4.
7) show^ that a closed^ subgeng H
of
" is^ moral
) K is^ a^ Galves^ extension^. solution. Let (^) H be closed and mesial. Then ~(i)=^ &Hst fro
all ~74. Thus ~(2)= L
are H &^ G.^ Hence^ 24Km Galas :
Conversely let / be^ Golois (^). Then och")=2"
fo all or (^) G (^).^ Is -(2) =^2 =^ asso↑ Stense (^) &(b) () =^ F=^ H^ = G(Y-+ )) = xxE! Thus H^ & (^) G (^).
⑤
- show^ that^ an open subject^ is normal in a iff will^ is^ a finite calous extension^. solution (^). Let (^) #4G be apar . By (6) (^) 4K is a
finite extension^
. Since HOG , /K^ is
alocs .
④ Let^ E be as intermediate subfield
I
bis .
show that the Kell
tapology on (^) G(2/E) is induced^ fro
the Errall
topology on (^3). Solution (^). Let H= (^) G1YE) (^) · Open sets in the
subspace topology^
elt one^ ton^ refuse
WC (^) - is (^) an oper
sat.^ We^ show^ that^ WiH
is (^) opens in the^ Kell topology m H^. Let 1 -> WhA- As^ U is (^) open 15 ⑦ 7 ↓ ata (^) e
- (^) -12/5) where (^) FIK is (^) a
finite
& -^ )s^ exte
H 3/45) =
SEEG
/- (^) fixes and^ += 2 on = [ = e4(4/E)(z = (^) 2m (^) + 3 => [ +^ =^ 3(4E)([= (^) 0m = (^) F 3
Stence 0 & (4/EF) G + &H . Note
k
that is a fixita Galins^ extension. I
EFE (^) can in Krall
topology
Hexat Ho^ W^ an (^) E f G(2/EC.^ /E
Let va^ G^ E^ I
/4) be^ pare^ in^ the^ -
we say & (^) ssume K Krall^ Palis topology e (^) Zer v (^) F %. 2e^ I^ -^ (r/E) <^ V fer · Finite Gabris extr^ FE
⑦ Since (^) &CK/IF(F Lim j (^) L-- G(k/(Fpe) r
2
the
isomrphism is (^) established (^). Since K
7 IF (^) , (K/Ep) = 3(k)k(x)) =T:
