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2011 | OriginalPaper | Chapter

Split Embedding Problems over Complete Fields

Author : Moshe Jarden

Published in: Algebraic Patching

Publisher: Springer Berlin Heidelberg

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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.

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Metadata
Title
Split Embedding Problems over Complete Fields
Author
Moshe Jarden
Copyright Year
2011
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-642-15128-6_7

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