7Local class field theory
III Local Fields
7.1 Infinite Galois theory
It turns out that the best way of formulating this theory is to not only use
finite extensions, but infinite extensions as well. So we need to begin with some
infinite Galois theory. We will mostly just state the relevant results instead of
proving them, because this is not a course on Galois theory.
In this section, we will work with any field.
Definition
(Separable and normal extensions)
.
Let
L/K
be an algebraic exten-
sion of fields. We say that
L/K
is separable if, for every
α ∈ L
, the minimal
polynomial
f
α
∈ K
[
α
] is separable. We say
L/K
is normal if
f
α
splits in
L
for
every α ∈ L.
Definition
(Galois extension)
.
Let
L/K
be an algebraic extension of fields.
Then it is Galois if it is normal and separable. If so, we write
Gal(L/K) = Aut
K
(L).
These are all the same definitions as in the finite case.
In finite Galois theory, the subgroups of
Gal
(
L/K
) match up with the
intermediate extensions, but this is no longer true in the infinite case. The
Galois group has too many subgroups. To fix this, we need to give
Gal
(
L/K
) a
topology, and talk about closed subgroups.
Definition
(Krull topology)
.
Let
M/K
be a Galois extension. We define the
Krull topology on M/K by the basis
{Gal(M/L) : L/K is finite}.
More explicitly, we say that
U ⊆ Gal
(
M/K
) is open if for every
σ ∈ U
, we can
find a finite subextension L/K of M/K such that σ Gal(M/L) ⊆ U.
Note that any open subgroup of a topological group is automatically closed,
but the converse does not hold.
Note that when
M/K
is finite, then the Krull topology is discrete, since we
can just take the finite subextension to be M itself.
Proposition.
Let
M/K
be a Galois extension. Then
Gal
(
M/K
) is compact
and Hausdorff, and if
U ⊆ Gal
(
M/K
) is an open subset such that 1
∈ U
, then
there is an open normal subgroup N ⊆ Gal(M/K) such that N ⊆ U.
Groups with properties in this proposition are known as profinite groups.
Proof. We will not prove the first part.
For the last part, note that by definition, there is a finite subextension of
M/K
such that
Gal
(
M/L
)
⊆ U
. We then let
L
0
be the Galois closure of
L
over
K
. Then
Gal
(
M/L
0
)
⊆ Gal
(
M/L
)
⊆ U
, and
Gal
(
M/L
0
) is open and normal.
Recall that we previously defined the inverse limit of a sequence rings. More
generally, we can define such an inverse limit for any sufficiently nice poset
of things. Here we are going to do it for topological groups (for those doing
Category Theory, this is the filtered limit of topological groups).
Definition
(Directed system)
.
Let
I
be a set with a partial order. We say that
I
is a directed system if for all
i, j ∈ I
, there is some
k ∈ I
such that
i ≤ k
and
j ≤ k.
Example. Any total order is a directed system.
Example. N with divisibility | as the partial order is a directed system.
Definition
(Inverse limit)
.
Let
I
be a directed system. An inverse system (of
topological groups) indexed by
I
is a collection of topological groups
G
i
for each
i ∈ I and continuous homomorphisms
f
ij
: G
j
→ G
i
for all i, j ∈ I such that i ≤ j, such that
f
ii
= id
G
i
and
f
ik
= f
ij
◦ f
jk
whenever i ≤ j ≤ k.
We define the inverse limit on the system (G
i
, f
ij
) to be
lim
←−
i∈I
G
i
=
(
(g
i
) ∈
Y
i∈I
G
i
: f
ij
(g
j
) = g
i
for all i ≤ j
)
⊆
Y
i∈I
g
i
,
which is a group under coordinate-wise multiplication and a topological space
under the subspace topology of the product topology on
Q
i∈I
G
i
. This makes
lim
←−
i∈I
G
i
into a topological group.
Proposition.
Let
M/K
be a Galois extension. The set
I
of finite Galois
subextensions
L/K
is a directed system under inclusion. If
L, L
0
∈ I
and
L ⊆ L
0
,
then we have a restriction map
·|
L
0
L
: Gal(L
0
/K) → Gal(L/K).
Then (Gal(L/K), ·|
L
0
L
) is an inverse system, and the map
Gal(M/K) → lim
←−
i∈I
Gal(L/K)
σ 7→ (σ|
L
)
i∈I
is an isomorphism of topological groups.
We now state the main theorem of Galois theory.
Theorem
(Fundamental theorem of Galois theory)
.
Let
M/K
be a Galois exten-
sion. Then the map
L 7→ Gal
(
M/L
) defines a bijection between subextensions
L/K
of
M/K
and closed subgroups of
Gal
(
M/K
), with inverse given by sending
H 7→ M
H
, the fixed field of H.
Moreover,
L/K
is finite if and only if
Gal
(
M/L
) is open, and
L/K
is Galois
iff Gal(M/L) is normal, and then
Gal(L/K)
Gal(M/L)
→ Gal(L/K)
is an isomorphism of topological groups.
Proof.
This follows easily from the fundamental theorem for finite field extensions.
We will only show that
Gal
(
M/L
) is closed and leave the rest as an exercise. We
can write
L =
[
L
0
⊆L
L
0
/K finite
L
0
.
Then we have
Gal(M/L) =
\
L
0
⊆L
L
0
/K finite
Gal(M/L
0
),
and each Gal(M/L
0
) is open, hence closed. So the whole thing is closed.