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2011 | OriginalPaper | Chapter
Algebraic Patching
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Let
E
be a field,
G
a finite group, and {
G
i
|
i
∈
I
} a finite set of subgroups of
G
with
G
=〈
G
i
|
i
∈
I
〉. For each
i
∈
I
we are given a Galois extension
F
i
of
E
with Galois group
G
i
. We suggest a general method how to ‘patch’ the given
F
i
’s into a Galois extension
F
with Galois group
G
(Lemma 1.1.7). Our method requires extra fields
P
i
, all contained in a common field
Q
and satisfying certain conditions making
$\mathcal{E}=(E,F_{i},P_{i},Q;G_{i},G)_{i\in I}$
into ‘patching data’ (Definition 1.1.1). The auxiliary fields
P
i
in this data substitute, in some sense, analytic fields in rigid patching and fields of formal power series in formal patching.
If in addition to the patching data,
E
is a Galois extension of a field
E
0
with Galois group Γ and Γ ‘acts properly’ (Definition 1.2.1) on the patching data
$\mathcal{E}$
, then we construct
F
above to be a Galois extension of
E
0
with Galois group isomorphic to Γ⋉
G
(Proposition 1.2.2).