itay kaplan and saharon shelah- automorphism towers and automorphism groups of fields without choice
TRANSCRIPT
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nlg
G G Aut(G)
Aut(G)
G | ord
G =
min { |G+1 = G }
G
G
G
D = x, y |x2 = y2 = 1
Aut(D) = D
G G
G < (2
)+
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norG (H) | ord H G G
nor0G (H) = H nor+1G (H) = norG (nor
G (H)) nor
G (H) =
{norG (H) | < }
G,H = min
nor+1G (H) = norG (H) .
nlg nlg > Aut(A),H
A H Aut(A)
nlg
nlg
+
ZF C
(2)+
> 2
G
G 1
nlg
K
H Aut(K) |K| = |H| =
P SL (2, K)
H
Aut(K)
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nlg|k
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n <
e = {s, t} l < N pnl+1
n <
(xs + xt + 1)
r = 0 K
char (K) = 2
r
|X| X X |X|
|X| |Y| X Y
|X| = |Y| X Y
G, H H G H G
A |A| ||A||
L
ZF C
X
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[X]
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k
nlg|k|
A ||A|| |k| Aut(A),H < H Aut(A) = G
|H| |k|
nlg
|k| = sup {G,H + 1}
G, H
k
nlg
|k| Aut(A),H
A, H
L
A
L
L
|L| |A||A| >
G G | ord
G0 = G
G+1 = Aut (G)
G
= {G
| < }
G
G = Inn(G) Aut(G) Aut(G)
G
Inn(G)
G G+1
G G = min { |G+1 = G } G
k |k| > G
G |k|
k
nlg
|k
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A
L
L
L |L| 0
|L| |A|
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S
G
Inn(S)
G Aut(S) G
G
Aut(S)
K
n <
GL (n, K) n n K
P GL (n, K) = GL (n, K) /Z(GL (n, K)) Z(GL (n, K)) KI
I
SL (n, K) = {x GL (n, K) |det (x) = 1 } Z(GL (n, K)) SL (n, K)
P SL (n, K) = SL (n, K) /Z(SL (n, K))
P SL (n, K)
P GL (2, K)
P SL (2, K)
K |K| 3
ZF
P
ZF C P
ZF P ZF P
V ZF V |= P
L = LV
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L
V
ZF C L |= P L |= ZF C
K
|K| 3 P SL (2, K)
L
4
H
L = {+, , 0, 1, H}
K 3 H K4
SL (2, K)
H
Z(SL (2, K)) Z(SL (2, K))
N
H
: H
Aut(N)
N H N H
N
H
K
P SL (2, K)
PL (2, K) := P GL (2, K)
Aut(K)
Aut(P SL (2, K)) = PL (2, K)
Aut(P SL (2, K)) Aut(K)
g P GL (2, K) x P SL (2, K) (x) = g (x) g1
K
Aut(P SL (2, K)) PL (2, K)
(, g)
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K
g P GL (2, K) Aut(K) x P SL (2, K)
(x) = gxg
1
(, g)
K
xt =
1 t
0 1
zt =
1 0
t 1
SL(2, K)
g P GL (2, K) Aut(P SL (2, K))
Aut(K) (x) = g (x) g1 x g1 (x) g xt x(t)
zt z(t) L 4
{i |i < 4 } K
P SL (2, K)
SL (2, K)
K4
Z(SL (2, K))
g P GL (2, K) t g1 (xt) g t g1 (zt) g
K
L
K
g P GL (2, K) t g1xtg t g1ztg
K
nlg
|k
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norPL(2,K) (G) = P GL (2, K)norAut(K) (H)
ZF C
K
Aut () = Aut(K) |K| = || + 0
= X, E K
|K| X
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|X+| X[
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i < N
+i = Xi, Ei
i x / ei
+i x ei
+i
|X|
{(1, x) |x X}
val (x)
val+ (1, x) = valG (x)val (x) rk (x) 0 val+ (2, u , w) =
valG (w) val+ (2, u, z) val+ (1, x)
Aut(+) (2, u, z) = (2, u, z) u E z
G
+{x,y} (2, u, z)
+u +u Aut(G) = {id,} w = x,y
(2, u , w) = (2, u, w)
= Aut() (x) = x
(1, x) = (1, x)
Aut (+)
Aut ()
F K F
K
x K\F F
p
x
F
p high
xi |i < F x0 = x xpi+1 = xi
F = Q
p
p high F 1, 1, 0
F = Fr r p (p,r 1) = 1 x xp
F
p high
r
0 p r {p0, . . . , pn1}
p,r
F
r
n
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k < n
Vk k = l Vk Vl = V =
k
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V |K|
F[
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ZF C
F
p
S
F
F (S, p)
{s (l) |s S, l < }
F
s (0) = s
s (l + 1)p = s (l) l <
S
p
s (l)
K1
= L (Y) (X, p0
)
Y = {x0t
|t X} L (Y)
L
l < N
Kl = Kl1 (El, pl+1)
El = {x0s + x
0t + 1 |{s, t} = e E, C(e) = l }
I
K KN1
0 1 K0 K1
0 1 i = Xi, Ei, Ci i < 2 X1 = X0 {t}
t / X0 K0 K1 x0t
z0 Tv X+ xs + 1
I
|X| 0 1 i = Xi, Ei, Ci i < 2
X1 = X0 {t} t / X0 I0 R = R0
Yt = {xit |i < } {x
ie |i < , t e E1 } It R [Yt]
t {e E1 |t e}
It
KN1
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: L [Y1 ] R [Yt] (I1) = It I0 =
ker ()
L[Y1]/I1 R[Yt]/It
Y X Y Y
Y = Y, E P(Y) RY = RY KY = KY
K
p a KX0 X0 X p high K a
p high KX0
i < {xis |s X} L
X1 X2 KX1 KX2 L
K
X1, X2
X2 = X1 {t} , t / X1
Aut ()
Aut(K)
: Aut()
Aut(K) () (xit) = x
i(t) () (x
ie) = x
i(e) Aut ()
t X, e E
() = id
(s) = t = s x0s = () (x0s) = x
0t
a K p high
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p = p0 a
{(xnss )ms |s X0, ms Z, ns < } X0 X p0 high
L
p = pl+1 l < N a
{(xnee )me |e E0, ne < , me Z} E0 E C E0 = l
pl+1 high L
X0 X a p high KX0
|X0| X0 =
X0 X1 X1 = X0 {t} t / X0
X0 a KX1 p high
e0 t X1 C(e0) = l
a KX0
e0 t X1 C(e0) = l
e0 = {s, t} , s X0 x0e0 = x0s + x0t + 1 KX1 KX0 x0t
x0t = x0e0 x
0s 1 r X0 er = {t, r}
x0er = x0e0 x
0s + x
0r X x
0s 1 X x
0s + x
0r
a
xie0m
c c
pl+1 high K0
z0 x0e0
e0 t X1 l t
1 s1, . . . , sk X0
C(si, t) = l k 2 X = X0\ {s1, . . . , sk} X = X {t}
|X| < |X1| KX X1
X s1, . . . , sk
t
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s X x0s p
p p0
X0 X
|X0|
Aut(K) {s0, t0} E l
{s1, t1} E x0s0 = x0s1 x0t0 = x0t1 : Aut () Aut(K)
{s1, t1} E l
x0s0
=x0s1m
x0t0
=
x0t1m
m r L
f1 =
x0s0
f2 =
x0t0
f =
x0s0 + x0t0 + 1
= f1 + f2 + 1
f1 = 1
(xiss )ms |s X0, ms Z, is <
f2 = 2
(xiss )ms |s Y0, ms Z, is <
f = 3
(xiee )me |e E0, me Z, ie <
X0, Y0 X E0 E E0 l
p = pl+1 f p high
is = 0
p0 ms
s X0 Y0
ie = 0
p me
e E0
f1+f2+1 = f pk
k = max {ie |e E0 }
1
xissms
+ 2
xittmt
+ 1pk
= pk
3
x0r + x
0w + 1
pki{r,w}m{r,w} .
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i = max {it |t X0 Y0} xitt (x
it)
piit0
xiss
x0r (xir)
pi0
x0w t T := X0 Y0
E0 yt = x
it
1
(ys)piis0 ms + 2
(yt)
piit0 mt + 1
pk= p
k
3
yp
i0
r + ypi0w + 1
pki{r,w}m{r,w}.
L (yt |t T)
mt m{r,w}
F
g F [X]
Z F(t) v (g (t)) = 1
v F = 0
v F [t] 0 v (m (t)) > 0 g|m m F [X]
g
mt0 t X0 Y0 v
L (yt |t T) v (yt0) = 1 v L (yt |t = t0 ) = 0
v (LHS) < 0
v (RHS) = 0
m{r,w} < 0 {r, w} E0 v L (yt |t T)
v (g (yr)) = 1 v L (yt |t = r ) = 0 g
Xpi0 + y
pi0w + 1 v
ypi0r + y
pi0w + 1
> 0
g
Xp
i0 + y
pi0w + 1
w = w v (RHS) < 0 v (yr) = 0
v (RHS) 0
T =
E0 = X0 Y0
L [yt |t T] pk
3 = 1
yr yw 0 r, w E0 = {{r, w}}
1y
piir0 mrr + 2y
piiw0 mww + 1
pk=
ypi0
r + ypi0w + 1
m{r,w}.
i = ir = iw 1 = 2 mr = mw i = 0 p0 mr
yr i = 0 k = 0 p m{r,w}
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f1 + f2 + 1 = 1 x0rmr + 2 x0w
mw + 1 = x0r + x0w + 1m{r,w} = f.
{r, w} l m := m{r,w} = mr 1 = 1 mr r
t X {t, t0} E (x0t ) =
(x0t)m
m
t X
t X (x0t ) = (x0t)
m
t
Aut()
(t) = t
pi L
(xit) =
xi(t)
m
(xie) =
xi(e)
m
m = 0
x0t r m = 0 = ()
Xi X i = 1, 2 KX1 KX2 =
KX1X2
X1, X2
x KX1 KX2
|X1|
x KX1 X1 X2 t X1\X2
X = X1\ {t} x / KX x KX
x0t KX (x) X2 = X
X2 X3 = X2 {t} x KX2 KX2
x0t KX3 KX2 x0t
KX (x) KX2 t X1 X2
: K X[
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p
K
Xp a a p K
n
K
r
r = 0
r n
n
0 = a K
z
Xn
= a
0 = b K(z)
bn
K
b = c zk
0 < k < n
c K
K t
Xp = a
a K L = K(t) b L bqm
K
q = p,m < b K
[L : K] = p
[L : K] = 1
[K(b) : K]
q
1
K
q
K
L
K L L K
K
L (y)
p
x K p high x L
x K\L y L (x)
xm m < pm
x
L (x) L (x1) L (x2) . . . K K/L (x)
L (xl) = L (xl+1) l xl+1 L (xl) xl p
L (xl)
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R F 1, . . . , n
F
1 F m1 (X1) R [X1]
2 F(1) m2 (1, X2) R [1, X2]
R [1, . . . , n] = R [X1, . . . , X n] / (m1, m2, . . . , mn) (m1, . . . , mn)
r
0
p
r {p0, . . . , pn1}
p,r
F
r
n k < n Vk k = l Vk Vl =
V = k
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1 j : V ( + 1\ {0})
F
zi, tvlv
|i < j, lv < (v) , v V
K
F(j,) = F(j,)Tv|vV
Xp zj1 F (j,) 1 j
w Vk Xpk tw(w)1 F(j,) , j
(w) <
k < n pmk 1 m F(j,)
F(zl) l < j
c f(zi)
g (zi)
vWk
tvrvlv
c F f g F i l Wk
Vk v Wk 1 rv < (v) rv m 0 < lv < prvk
F
K
K
F(j,)
F
S
K
F(j,)
R/I
R = S[Yi, Svl |i, l < ,v V ] R = S
Yi, S
vlv |i < j, lv < (v) , v V
I R
Ypi+1 = Yi i < i < j
Yv0 = Tv (Y0) v V
v Vk k < n Svl+1
pk = Svl l < l < (v)
s high K F s p
pk |k < n
p high K c (zi)m
c p high
F
i <
m
p
p
z0 p
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V |K|
F[ 1
: V
( + 1\ {0}) supp () = {v | (v) = 1 } || =
{ (v) 1 |v supp () , (v) < }
supp () || [V]
supp ()
j
j = 1
i <
Siv F [X] Siv (X) = Tv
Xp
i
v V {Siv |v V }
j
supp ()
F(j,)Tv |vV = F(1, )Sj1v |nV
zj1 z0 Tv Sj1v j = 1
j
x F(i + 1, )
xqm
F(i, ) x F (i, ) j = l + 1
j = 1
|| || = 0 F(1, ) =
F(z0) F [z0]
tw(w) / F(1, )
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g (z0) tw(w) = cf(z0)
vW
tvrvlv
W V 1 rv (w) 0 < lv < qrv
rv < (v) v W
q
(w)
gq(w)
Tw = cq(w)fq
(w)vW
Tq
(w)rv
v
lv
Tv g = f = 1
W = {w} rw = (w)
b F(1, ) bqm
F(z0) m > 0
v V (v) > 1
(v) = (v) m (v) m < 1 (v) = 1
(w) = (w) w = v
F(1, )
tv(v)1
= F(1, ) b F(1, )
tv(v)1
bq
m
F (z0) L := F(1, )
tv(v)1
qm L b = c
tv(v)1
l
c L
0 < l < qm q | l q l
(v) 1 m cqm
F(z0)
c tv
(v) = 1
(v) 1 > m
cq(v)1
F(z0) c L c
d f(z0)
g (z0)
uW
turulu
d F W V 1 ru < (u) , ru (v) 1 u W
W supp () c cqm
F(1, ) (w) =
(w) m w V (w) m < 1 (w) = 1 bqm
F(z0) tv(v)1
lqm=
tv(v)m1
l F(1, ) q l (l, qm) = 1 tv(v)m1 F(1,
)
(v) (v) 1 = (v) m 1
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tv(v) F(1, )
j = 1
||
||
z1 / F(1, ) || = 0
|| x K
F
L = F(x)
z0 L x
L
F
v V
+ : V + 1 + (v) = (v) + 1 w = v + (w) = (w)
x F(, +) \F(, ) F(, ) (x) = F(, +) [F(, +) : F(, )] = q
tv(v) F(, ) (x) L (, )
x F(, 1) 1 x F(zi)
x F
S(j,)
I(j,)
R
I
S(j,) /I(j,) K
F (j,)
j, || R = S[z0] = S[Y0]
K R
zi, tvlv |i < j, lv < (v)
=
S[Y0]
Yi, Svlv |1 < i < j, lv < (v)
/I(j,) = S(j,) /I(j,)
x
s high K x F(i, ) i <
|| x s high F(i, )
x F(, ) p high x p high
F(, 0) |0| || p high
F(, )
c zmi
x F(, 1)
x
p high F (zi) i x F
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x
p high i < x F(zi)
f0, g0 F [zi] zi
x = z
l0
i
f0(zi)
g0(zi)
l0 Z
u = x/z
l0
i
p high
F (zi) Xm =
y F(, 1)ypm = u
j <
Xj F(zi) , Xj+1 F(zi) s
Xj+1 F(zs) s > i v Xj+1 v = vp Xj F(zi) v
= zl1if1(zi)g1(zi)
f1, g1 zi l1 Z (v)pj = u
l1 = 0 vp F (zi) v = zms d d F(zi) m < p m
v / F(zs) s < s d = (zi)l2 f2(zi)
g2(zi)
f2, g2 F [zi]
zi l2 Z vp = v
zms1 zl2pi
f2 (zi)
g2 (zi)
p=
f1 (zi)
g1 (zi)
p
s 1 i psil2 + m = 0 p | m
+
+ (v) = (v) + 1
v V + (w) = (w) w = v x F(, +) p high
K = F(, +)
L = F(, )
K/L
q
N : K L
N(a) = aq a L
Ki = F (i, +) Li = F(i, ) N Ki = NKi : Ki Li
N(x)
p high L y = xq/N(x) p high K i <
x, y F(i + 1, +) y p high F(i + 1, +)
u F(, +) \F(i + 1, +) up F (i + 1, +) y pm u
m <
k = max {n |u / F (n + 1, +) } i u F(k + 2, +)
up F(k + 1, +) u = h (zk+1)b
h F(k + 1, +) 0 < b