Ample Fields

One of the major problems of Field Arithmetic was whether the absolute Galois group of every countable PAC Hilbertian field

K

is free of countable rank. By Iwasawa, that means that every finite embedding problem of Gal(

K

) is solvable. The PAC property of

K

implies that Gal(

K

) is projective, so it suffices to solve finite split embedding problems over

K

. Since

K

is Hilbertian, it suffices to solve finite split constant embedding problems over

K

(

x

), where

x

is transcendental over

K

. Since

K

is PAC, it is existentially closed in the field of formal power series

$\hat{K}=K((t))$

. By Bertini-Noether, it suffices to solve each finite split constant embedding problem over

$\hat{K}(x)$

. Thus, the initial problem of proving that

$\mathrm{Gal}(K)\cong \hat{F}_{\omega}$

is reduced to a problem that Proposition 4.4.2 settles.

The property of being existentially closed in

K

((

t

)) that each PAC field

K

has is shared by all Henselian fields. We call a field

K

which is existentially closed in

K

((

t

))

ample

. In that case, the arguments of the preceding paragraph prove that each finite split constant embedding problem over

K

(

x

) is solvable (Theorem 5.9.2).

It turns out that ample fields can be characterized in diophantine terms: A field

K

is ample if and only if every absolutely irreducible curve over

K

with a simple

K

-rational point has infinitely many

K

-rational points (Lemma 5.3.1). Surprisingly enough, each field

K

such that Gal(

K

) is a pro-

p

group for a single prime number

p

has the latter property and is therefore ample (Theorem 5.8.3). On the other hand, the theorems of Faltings and Grauert-Manin imply that number fields and function fields of several variables are not ample (Proposition 6.2.5).