20 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN

Use (5) and equation

(3)

of Section

1

to translate

(1)-(3)

via

f

from

1i

to

H'.

Let

qj

=

f-

1

(Q.;)

E

H'(K),

for j = 1, ...

,m.

Then,

(1')

qj

E

H'(K(t)),

for each

t

E

Xj,

for j =

1, ... ,

m.

We must find p'

E

H'(K)

such that

(2')

p'

E H'(K(a))

for each

a E G(K),

that is, p'

E H'(K);

(3')

liP'-

qjll~ ~

v2

III/II~

in

K(t),

for each

t

E

Xj,

for j =

1, ... ,

m.

Suppose

t

E

Xj

for some 1

~

j

~

m.

As

11/11~

is algebraic over

K,

there

is

aj

E

K

with

a] II/II~

in

K(t).

Replace

xl, ...

'Xm

by a finer partition to

assume this is true for each

t E

Xi.

Thus

(3')

follows from a stronger statement:

(3")

liP'-

qjll~ ~ v

2

la]

in

K(t),

for each

t

E

Xj,

for j =

1, ... ,m.

By Proposition 1.2 there is p' E

H'(K)

such that (3") holds. o

THEOREM

5.2.

Let Ll K be a finite Galois extension with L not formally

real

and

let

rr:

H - G(LI K) be

an

epimorphism of finite groups. For each

1

~

j

~

m

let Ij

~

H

be

a conjugacy

domain of involutions such that rr(Ij)

=

I xi ( L I K). Then there exists

a

regular extension F of L, Galois over K ( x),

and an

isomorphism h1 :G(FIK(x))- H that maps Ix;(FIK(x)) onto Ij· In

addition, the following diagram commutes.

G ( F

I

K (X))

_----'.::h.!..._l

--+

H

r~

/

G(LIK)

In particular,

h1

maps I(FIK(x)) onto

Uj

Ij·

PROOF. By Skolem-Lowenheim Principle

[FJ,

Proposition 6.4] we may as-

sume that K

~

C. We divide the proof into five parts.

Part

1.

Weakening of commutativity.

Let

G

= Ker(

1r).

Instead of commuta-

tivity of the diagram it suffices to show that h1 maps

G(FI L(x))

ont~

G.

Indeed,

apply Lemma

3.3.

This gives an epimorphism of finite groups p:

H - H

and

conjugacy domains of involutions

h, ... ,

fm ~

ii

with

p(f;)

=

Ij.

In addition,

every automorphism of

G(LI K)

that preserves the

Ix; (LI K)

lifts (under p orr)

to an automorphism of

ii

that preserves the

'4.

Let

G

= Ker(p orr).

Assume that we can find a regular extension

P

of

L,

Galois over

K(x),

and an isomorphism

h1

:G(FIK(x)) -

ii

that maps

G(FIL(x))

onto

G

and

the

Ix;(FIK(x))

onto the

fj.

In particular, Ker(rr o po

h1

)

=

G(F/L(x))

=

Ker(resL). Hence there exists an automorphism a of

G(L/ K)

such that a o

1r

o

p

o

h1

=res£ and a preserves the

Ix; (L/ K).

We can lift a to an automorphism

a

of

ii

that preserves the

ij.

Thus, by composing h

1

with

a

we may assume

that (rr o

p)

o

h1

=res£.

Now let F

be

the fixed field of Ker(p) in

P.

Then h1 induces an isomorphism

h

1

:

G(F/ K(x))- H

with the required properties.