Itay Kaplan and Saharon Shelah- Automorphism Towers and Automorphism Groups of Fields Without Choice

8/3/2019 Itay Kaplan and Saharon Shelah- Automorphism Towers and Automorphism Groups of Fields Without Choice

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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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modified:2011-08-03

nlg|k


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modified:2011-08-03

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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modified:2011-08-03

|X+| X[


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modified:2011-08-03

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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modified:2011-08-03

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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modified:2011-08-03

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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modified:2011-08-03

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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modified:2011-08-03

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

Documents

Itay Kaplan and Saharon Shelah- Automorphism Towers and Automorphism Groups of Fields Without Choice