7Local class field theory

III Local Fields 7.2 Unramified extensions and Weil group
We first define what it means for an infinite extension to be unramified or totally
ramified. To do so, we unexcitingly patch up the definitions for finite cases.
Definition
(Unramified extension)
.
Let
K
be a local field, and
M/K
be alge-
braic. Then
M/K
is unramified if
L/K
is unramified for every finite subextension
L/K of M/K.
Note that since the extension is not necessarily finite, in general
M
will not
be a local field, since chances are its residue field would be infinite.
Definition
(Totally ramified extension)
.
Let
K
be a local field, and
M/K
be
algebraic. Then
M/K
is totally ramified if
L/K
is totally ramified for every
finite subextension L/K of M/K.
Proposition.
Let
M/K
be an unramified extension of local fields. Then
M/K
is Galois, and
Gal(M/K)
=
Gal(k
M
/k
K
)
via the reduction map.
Proof.
Every finite subextension of
M/K
is unramified, so in particular is Galois.
So
M/K
is Galois (because normality and separability is checked for each
element). Then we have a commutative diagram
Gal(M/K) Gal(k
M
/k
K
)
lim
L/K
Gal(L/K) lim
L/K
Gal(k
L
/k
K
)
reduction
reduction
The left hand map is an isomorphism by (infinite) Galois theory, and since all
finite subextensions of
k
M
/k
K
are of the form
k
L
/k
K
by our finite theory, we
know the right-hand map is an isomorphism. The bottom map is an isomorphism
since it is an isomorphism in each component. So the top map must be an
isomorphism.
Since the compositor of unramified extensions is unramified, it follows that
any algebraic extension M/K has a maximal unramified subextension
T = T
M/K
/K.
In particular, every field K has a maximal unramified extension K
ur
.
We now try to understand unramified extensions. For a finite unramified
extension L/K, we have an isomorphism
Gal(L/K) Gal(k
L
/k
K
)
,
By general field theory, we know that
Gal
(
k
L
/k
K
) is a cyclic group generated by
Frob
L/K
: x 7→ x
q
,
where
q
=
|k
K
|
is the size of
k
K
. So by the isomorphism, we obtain a generator
of Gal(L/K).
Definition
(Arithmetic Frobenius)
.
Let
L/K
be a finite unramified extension
of local fields, the (arithmetic) Frobenius of
L/K
is the lift of
Frob
L/K
Gal(k
L
/k
K
) under the isomorphism Gal(L/K)
=
Gal(k
L
/k
K
).
There is also a geometric Frobenius, which is its inverse, but we will not use
that in this course.
We know
Frob
is compatible in towers, i.e. if
M/L/K
is a tower of finite
unramified extension of local fields, then
Frob
M/K
|
L
=
Frob
L/K
, since they both
reduce to the map
x 7→ x
|k
K
|
in
Gal
(
k
L
/k
K
), and the map between
Gal
(
k
L
/k
K
)
and Gal(L/K) is a bijection.
So if M/K is an arbitrary unramified extension, then we have an element
(Frob
L/K
) lim
L/K
Gal(L/K)
=
Gal(M/K).
So we get an element
Frob
M/K
Gal
(
M/K
). By tracing through the proof of
Gal(M/K)
=
Gal(k
M
/k
K
), we see that this is the unique lift of x 7→ x
|k
K
|
.
Note that while for finite unramified extensions
M/K
, the Galois group is
generated by the Frobenius, this is not necessarily the case when the extension
is infinite. However, powers of the Frobenius are the only things we want to
think about, so we make the following definition:
Definition
(Weil group)
.
Let
K
be a local field and
M/K
be Galois. Let
T
=
T
M/K
be the maximal unramified subextension of
M/K
. The Weil group
of M/K is
W (M/K) = {σ Gal(M/K) : σ|
T
= Frob
n
T/K
for some n Z}.
We define a topology on
W
(
M/K
) by saying that
U
is open iff there is a finite
extension L/T such that σ Gal(L/T ) U.
In particular, if M/K is unramified, then W (M/K) = Frob
Z
T/K
.
It is helpful to put these groups into a diagram of topological groups to see
what is going on.
Gal(M/T ) W (M/K) Frob
Z
T/K
Gal(M/T ) Gal(M/K) Gal(T/K)
Here we put the discrete topology on the subgroup generated by the Frobenius.
The topology of
W
(
M/K
) is then chosen so that all these maps are continuous
homomorphisms of groups.
In many ways, the Weil group works rather like the Galois group.
Proposition.
Let
K
be a local field, and
M/K
Galois. Then
W
(
M/K
) is dense
in
Gal
(
M/K
). Equivalently, for any finite Galois subextension
L/K
of
M/K
,
the restriction map W (M/K) Gal(L/K) is surjective.
If L/K is a finite subextension of M/K, then
W (M/L) = W (M/K) Gal(M/L).
If L/K is also Galois, then
W (M/K)
W (M/L)
=
Gal(L/K)
via restriction.
Proof.
We first prove density. To see that density is equivalent to
W
(
M/K
)
Gal
(
L/K
) being surjective for all finite subextension
L/K
, note that by the
topology on
Gal
(
M/K
), we know density is equivalent to saying that
W
(
M/K
)
hits every coset of
Gal
(
M/L
), which means that
W
(
M/K
)
Gal
(
L/K
) is
surjective.
Let
L/K
be a subextension. We let
T
=
T
M/K
. Then
T
L/K
=
T L
. Then
we have a diagram
Gal(M/T ) W (M/K) Frob
Z
T/K
Gal(L/T L) Gal(L/K) Gal(T L/K)
Here the surjectivity of the left vertical arrow comes from field theory, and the
right hand vertical map is surjective because
T L/K
is finite and hence the
Galois group is generated by the Frobenius. Since the top and bottom rows are
short exact sequences (top by definition, bottom by Galois theory), by diagram
chasing (half of the five lemma), we get surjectivity in the middle.
To prove the second part, we again let
L/K
be a finite subextension. Then
L · T
M/K
T
M/L
. We then have maps
Frob
Z
T
M/K
/K
Gal(T
M/K
/K) Gal(k
M
/k
K
)
Frob
Z
T
M/L
/L
Gal(T
M/L
/L) Gal(k
M
/k
L
)
=
=
So the left hand vertical map is an inclusion. So we know
Frob
Z
T
M/L
/L
= Frob
Z
T
M/K
/K
Gal(T
M/L
/L).
Now if σ Gal(M/L), then we have
σ W (M/L) σ|
T
M/L
/L
Frob
Z
T
M/L
/L
σ|
T
M/K
/K
Frob
Z
T
M/K
/K
σ W (M/K).
So this gives the second part.
Now
L/K
is Galois as well. Then
Gal
(
M/L
) is normal in
Gal
(
M/K
). So
W (M/L) is normal in W (M/K) by the second part. Then we can compute
W (M/K)
W (M/L)
=
W (M/K)
W (M/K) Gal(M/L)
=
W (M/K) Gal(M/L)
Gal(M/L)
=
Gal(M/K)
Gal(M/L)
=
Gal(L/K).
The only non-trivial part in this chain is the assertion that
W
(
M/K
)
Gal
(
M/L
) =
Gal
(
M/K
), i.e. that
W
(
M/K
) hits every coset of
Gal
(
M/L
), which is what
density tells us.