7Local class field theory

III Local Fields 7.3 Main theorems of local class field theory
We now come to the main theorems of local class field theory.
Definition
(Abelian extension)
.
Let
K
be a local field. A Galois extension
L/K is abelian if Gal(L/K) is abelian.
We will fix an algebraic closure
¯
K
of
K
, and all algebraic extensions we will
consider will be taken to be subextensions of
¯
K/K
. We let
K
sep
be the separable
closure of K inside
¯
K.
If
M/K
and
M/K
are Galois extensions, then
LM/K
is Galois, and the map
given by restriction
Gal(LM/K) Gal(L/K) × Gal(M/K).
is an injection. In particular, if
L/K
and
M/K
are both abelian, then so is
LM/K. This implies that there is a maximal abelian extension K
ab
.
Finally, note that we know an example of an abelian extension, namely the
maximal unramified extension
K
ur
=
T
K
sep
/K
K
ab
, and we put
Frob
K
=
Frob
K
ur
/K
.
Theorem
(Local Artin reciprocity)
.
There exists a unique topological isomor-
phism
Art
K
: K
×
W (K
ab
/K)
characterized by the properties
(i) Art
K
(π
K
)|
K
ur
= Frob
K
, where π
K
is any uniformizer.
(ii) We have
Art
K
(N
L/K
(x))|
L
= id
L
for all L/K finite abelian and x L
×
.
Moreover, if
M/K
is finite, then for all
x M
×
, we know
Art
M
(
x
) is an
automorphism of
M
ab
/M
, and restricts to an automorphisms of
K
ab
/K
. Then
we have
Art
M
(x)|
K
ab
K
= Art
K
(N
M/K
(x)).
Moreover, Art
K
induces an isomorphism
K
×
N
M/K
(M
×
)
Gal
M K
ab
K
.
To simplify this, we will write
N
(
L/K
) =
N
L/K
(
L
×
) for
L/K
finite. From
this theorem, we can deduce a lot of more precise statements.
Corollary. Let L/K be finite. Then N(L/K) = N((L K
ab
)/K), and
(K
×
: N(L/K)) [L : K]
with equality iff L/K is abelian.
Proof.
To see this, we let
M
=
L K
ab
. Applying the isomorphism twice gives
K
×
N(L/K)
=
Gal(M/K)
=
K
×
N(M/K)
.
Since
N
(
L/K
)
N
(
M/K
), and [
L
:
K
]
[
M
:
K
] =
|Gal
(
M/K
)
|
, we are
done.
The theorem tells us if we have a finite abelian extension
M/K
, then we
obtain an open finite-index subgroup
N
M/K
(
M
×
)
K
×
. Conversely, if we are
given an open finite index subgroup of
K
×
, we might ask if there is an abelian
extension of
K
whose norm group is corresponds to this subgroup. The following
theorem tells us this is the case:
Theorem. Let K be a local field. Then there is an isomorphism of posets
open finite index
subgroups of K
×
finite abelian
extensions of L/K
H (K
ab
)
Art
K
(H)
N(L/K) L/K
.
In particular, for L/K and M/K finite abelian extensions, we have
N(LM/K) = N(L/K) N(M/K),
N(L M/K) = N (L/K)N(M/K).
While proving this requires quite a bit of work, a small part of it follows from
local Artin reciprocity:
Theorem.
Let
L/K
be a finite extension, and
M/K
abelian. Then
N
(
L/K
)
N(M/K) iff M L.
Proof.
By the previous theorem, we may wlog
L/K
abelian by replacing with
L K
ab
. The direction is clear by the last part of Artin reciprocity.
For the other direction, we assume that we have
N
(
L/K
)
N
(
M/K
), and
let
σ Gal
(
K
ab
/L
). We want to show that
σ|
M
=
id
M
. This would then imply
that M is a subfield of L by Galois theory.
We know
W
(
K
ab
/L
) is dense in
Gal
(
K
ab
/L
). So it suffices to show this for
σ W (K
ab
/L). Then we have
W (K
ab
/L)
=
Art
K
(N(L/K)) Art
K
(N(M/K)).
So we can find
x M
×
such that
σ
=
Art
K
(
N
M/K
(
x
)). So
σ|
M
=
id
M
by local
Artin reciprocity.
Side note: Why is this called “class field theory”? Usually, we call the field
corresponding to the subgroup
H
the class field of
H
. Historically, the first type
of theorems like this are proved for number fields. The groups that appear on
the left would be different, but in some cases, they are the class group of the
number field.