Constant Split Embedding Problems over Complete Fields

Let

K

0

be a complete field under a discrete ultrametric absolute value and

x

an indeterminate. We prove that each finite split embedding problem over

K

0

has a

rational solution

. Thus, given a finite Galois extension

K

of

K

0

with Galois group Γ that acts on a finite group

G

, there is a finite Galois extension

F

of

K

0

(

x

) which contains

K

(

x

) with Gal(

F

/

K

(

x

))≅

G

and Gal(

F

/

K

0

(

x

))≅Γ⋉

G

such that res: Gal(

F

/

K

0

(

x

))→Gal(

K

/

K

0

) corresponds to the projection Γ⋉

G

→Γ. Moreover,

F

has a

K

-rational place unramified over

K

(

x

) whose decomposition group over

K

0

(

x

) is Γ.

To construct

F

we choose finitely many cyclic subgroups

C

i

,

i

∈

I

, of

G

which generate

G

. For each

i

∈

I

we construct a Galois extension

F

i

=

K

(

x

,

z

i

) of

K

(

x

) with Galois group

C

i

in

K

((

x

)). Then we consider the ring

R

=

K

{

w

i

|

i

∈

I

} as in Section 3.2, where

$w_{i}={r\over x-c_{i}}$

,

r

∈

K

0

,

c

i

∈

K

, and |

r

|≤|

c

i

−

c

j

| for all

i

≠

j

, and shift

F

i

into the field

P

′

i

=Quot(

K

{

w

i

}) (Lemma 4.3.5). Choosing the

c

i

’s in an appropriate way (Claim A of the proof of Proposition 4.4.2), we establish patching data

$\mathcal{E}$

with a proper action of Γ and apply Proposition 1.2.2 to solve the given embedding problem.