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

Observe that

F

is the maximal extension of

JR(x)

unramified outside the primes

of

IR(x)

induced by ai, ... , ar,

oo.

In this situation the proof of

[KN,

Satz

2]

shows that there is f.

E

fi

=

a(F jlR(x))

such that €, cTI, ... ,

cTn

form a system

of generators for

fi

with the defining relations

(10)

€2

=

1

and

for 1 :::; j :::;

e.

Further, u!+j = a-;~(

2

m+I-j) for j = 1, ... , 2m. By (3) and (4) this implies that

resp€ and

f.

act on

a

in the same way. Since H is a subgroup of Aut(a), this

implies that resp€

=E.

Each involution of

fi

is conjugate to one of €, cTI €, u2ui €, ... ,

u

e · · ·

u2ui €.

Indeed, by

[HJ2,

Lemma 4.2 (Part E)],

fi

is the free profinite product of the

free profinite group (ue+I, ... ,

un)

of rank

n- e

=

m

withe+ 1 groups

(€), (a-If.), (a-2a-I€), ... , (a-e · · · u2ui€)

that are of order two. Thus by

[HR,

Theorem A'] the elements of finite order in

fi

are the conjugates of the elements of these e + 1 subgroups.

By Lemma

4.5

below, all involutions of

fi

are real. Using (9) and

(5)

conclude

I(F/IR(x))

=

respl(F

/IR(x))

=

respConif( {€, cTIE, ... ,

cTe · · ·

ui€})

=

=

ConH( {

E, O"If., ... ,

O"e • • •

O"If.})

=

ConH(I).

o

LEMMA

4.5.

Let S be

a

finite set of finite prime divisors of the field

JR(x).

Let IR(x)s be the maximal extension of!R(x) unramified outsideS

U {oo},

and

set as= a(JR(x)s jlR(x)). Then all involutions of as

are

real.

PROOF. By

[KN,

Satz

3]

the absolute Galois group

g

of

IR(x)

has generators

{o,

Tp

I

p

a finite prime of

JR(x)/IR}

with the defining profinite relations

82 =

1 and r:

= (

II

r;~)

r;I (

II

r;~)

-I for all real

p.

p 'p p 'p

Here

TIP

'p r;~ is the unique accumulation point of the products r;~ · · · r;~ E

g

for real primes PI, ... , Pr with PI

· · ·

Pr

p.

Furthermore,

[KN]

constructs this system of generators in such a way that for every finite set

S

of

finite primes and every finite prime

p

f/.

S

the natural restriction map

g

~

as

maps

Tp

onto

1 [KN,

p.

207].

Let PI

· · ·

Pe be the real, and Pe+I, ... , Pn the complex primes of

S.

Let

cTj

be the image of

Tp

j,

for j

=

1, ... ,

n,

and let E be the image of

0

in

as.

Then

€, cTI, ... ,

cTn

generate

as

and satisfy (10). These are in fact defining relations

for

as

by

[KN,

Satz

2].

As in the last part of the proof of Proposition 4.4,

each involution of

as

is conjugate to some

cTj · • ·

ui€, where 0:::; j :::;

e.

Thus it

suffices to show that each

cTj · · ·

cTIE lifts to an involution of

9.

To this end put

Oo=O and Oj=(

II

r;~)-Io=rp 1 (

II

r;~)-Io

p

'~p j

p 'p

j