2011 | OriginalPaper | Chapter
Split Embedding Problems over Complete Fields
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Let
K
0
be a complete field with respect to an ultrametric absolute value. In Proposition 4.4.2 we considered a finite Galois extension
K
of
K
0
with Galois group Γ acting on a finite group
G
and let
x
be an indeterminate. We constructed a finite Galois extension
F
of
K
0
(
x
) that contains
K
and with Galois group Γ⋉
G
that solves the constant embedding problem Γ⋉
G
→Gal(
K
(
x
)/
K
0
(
x
)). Using an appropriate specialization we have been then able to prove the same result in the case where
K
0
was an arbitrary ample field (Theorem 5.9.2). This was sufficient for the proof that each Hilbertian PAC field is
ω
-free (Theorem 5.10.3).
In this chapter we lay the foundation to the proof of the third major result of this book: Giving a function field
E
of one variable over an ample field
K
of cardinality
m
, each finite split embedding problem over
E
has
m
linearly disjoint solution fields (Theorem 11.7.1).
Here we let
K
0
be as in the first paragraph, and consider a finite Galois extension
E
′ of
K
0
(
x
) (where
E
′ is not necessarily of the form
K
(
x
) with
K
/
K
0
Galois) acting on a finite group
H
. We prove that the finite split embedding problem Gal(
E
′/
K
0
(
x
))⋉
H
→Gal(
E
′/
K
0
(
x
)) has a solution field
F
′. Moreover, if
H
is generated by finitely many cyclic subgroups
G
j
, then for each
j
there is a branch point
b
j
with
G
j
as an inertia group.