These are my live-TeXed notes for the course *Math 223a: Algebraic Number Theory* taught by Joe Rabinoff at Harvard, Fall 2012.

09/05/2012

##
Introduction

This is a one-year course on class field theory — one huge piece of intellectual work in the 20th century.

Recall that a *global field* is either a finite extension of (characteristic 0) or a field of rational functions on a projective curve over a field of characteristic (i.e., finite extensions of ). A *local field* is either a finite extension of (characteristic 0) or a finite extension of (and sometimes we also include and as local fields) . The major goal of class field theory is to describe all abelian extensions of local and global fields (an *abelian extension* means a Galois extension with an abelian Galois group). Suppose is the maximal abelian extension of , then , the topological abelianization of the absolute Galois group . Moreover, there is a bijection between abelian extensions of and closed subgroups of . So we would like to understand the structure of .

We also would like to know information about ramification of abelian extensions. For example, does have a degree 3 extension ramified only over 5? This can be nicely answered by class field theory. Class field theory also allows us to classify *infinite* abelian extensions via studying the topological group . The course will start with lots of topological groups in the first week and one may be impressed by how seemingly unrelated to number theory at first glimpse.

Here some useful applications of class field theory.

(Primes in arithmetic progressions)
The famous Dirichlet theorem says that for an integer

, the primes

is equidistributed in

. Notice that there is a canonical isomorphism between

and

and this isomorphism sends the Frobenius element associated to any

to

. So the classical theorem of Dirichlet can be viewed as a special case of the following Chebotarev's density theorem.

Let

be a finite Galois extension. Then the Frobenius elements (conjugacy classes)

for primes

of

are equidistributed on the conjugate classes of

.

Chebotarev's density theorem is proved via reducing to the case of cyclic extensions using a nice counting argument and then applying class field theory (cf. 34).

(Artin -functions)
An

*Artin representation* is a continuous representation

where

is a number field and

is a finite complex vector space. One can attach an

*Artin -function* to each Artin representation. When

is one-dimensional, an Artin representation is simply a character of

, which must factor through

. So a one-dimensional Artin representation is nothing but a character of

. By continuity, this character factors through

, where

is a finite abelian extension of

.

Weber generalized the Dirichlet -functions to Weber -functions over any number fields. He proved that a Weber -function has analytic continuation to the whole and satisfies a functional equation. Class field theory then tells us that the Weber -functions are exactly the one-dimensional Artin -functions.

Let

be a smooth projective connected curve over a finite field and

be the function field of

. Then class field theory classifies all abelian covers of

. In particular, this gives the abelianization of the etale fundamental group

of

. The proof of Weil conjecture II uses it in an essential way.

(Cohomology of )
In the second semester we will study the Tate global and local duality, Brauer groups and introduce all the cohomological machinery in order to prove class field theory.

Now we briefly turn to the main statements of class field theory. Class field theory gives Artin maps (in the global case) and and the kernel and image of the Artin maps can be described. The crucial thing is that the source of the Artin maps are intrinsic to the field (doesn't involve ). Moreover the Artin maps satisfy the local-global compatibility: the diagram commutes. In other words, class field theory is *functorial* in . For a finite abelian extension , the Artin map induces the relative Artin maps and . They are surjective and the kernel is exactly the norm subgroups. We can furthermore read ramification data from the relative Artin maps. In the local case is exactly the inertia group and is exactly the -th ramification group of . In the global case, the ramification data can be extracted from the local-global compatibility. We will spend most of the first semester to state the class field theory and draw important consequences from it and devote the second semester to the proofs.

Here are a few words about the proofs of class field theory. The classical approach is to do the global case first, using cyclotomic extensions, Kummer extensions and Artin-Schreier extensions (in characteristic ) to fill up the absolute Galois group, and then derive the local case from it. The cohomological approach is to establish local class field theory using group cohomology and then "glue" the local Artin maps to obtain the global Artin maps. One of the advantage of the cohomological approach is that the local-global compatibility comes from the construction. We will take this approach in the second semester.

Finally we may also talk about explicit class field theory, i.e., finding explicit construction (e.g. as splitting field of polynomials) of abelian extensions. This is a highly open problem in general with several known cases:

- When , , the union of all cyclotomic extensions of . So the polynomials exhaust all abelian extensions of . This the most satisfactory case.
- When is an imaginary quadratic field. The CM theory of elliptic curves assert that can be obtained essentially by adjoining all the torsion points on an elliptic curve with complex multiplication by .
- When is a global function field, there is a theory of Drinfeld modules to obtain most abelian extensions of (apart from some ramification restriction).
- When is a nonarchimedean local field, Lubin-Tate theory tells that can be obtained by adjoining all torsion points of the Lubin-Tate formal groups.

Somehow adjoining torsion points of a group law is possibly the only known way to construct explicit class fields.

09/07/2012

##
Global fields

Today and next Monday we will review the basic notions we learned from Math 129, taking this opportunity to set up the notations.

A

*number field* is a finite extension of

. It is an abstract field and have many embeddings into the complex numbers

(we do not specify one).

A

*global function field* is a finite (separable) extension of

. It is a fact that the algebraic closure

of

in a global function field

is a finite field, called the

*field of constants*. Equivalently,

is the field of rational functions on a smooth projective geometrically connected curve

over

, unique up to isomorphism. The geometrically connectedness ensures that

is the field of constants.

A *global field* is either a number field or a global function field.

Let

be a global field. A

*place* of

is an equivalent class

of nontrivial absolute values

on

. Two absolute values

and

are equivalent if and only if they induce the same topology on

under the metric

, if and only if

for

. The set of all places of

is denoted by

. Suppose

, then we have the

*completion* of

with respect to any absolute values

corresponding to

.

When

, there is only one archimedean place

given by the usual absolute value and

. For any prime number

, the

-adic absolute value

is defined on the generators of

by

In this way we have a bijection between the set of primes of

and all non-archimedean places of

. The completion with respect to

is denoted by

.

Suppose

is an archimedean place of

, then there exists an absolute value

such that

for any

. By the Gelfand-Mazur theorem,

is either

or

, so the absolute value

is the usual absolute value on

or

. The

*normalized absolute value* is often defined as

(when

is real) and

(when

is complex). This normalization simplifies many statements like the product formula.

Suppose

is a non-archimedean place of

. Let

be a representative of

. Then

is discrete (Ostrowski's theorem), hence is equal to

for some

. We normalized the valuation

such that

. This valuation is intrinsic to

and is called the

*normalized valuation*. It extends uniquely to

.

We denote

, the

*valuation ring* of

;

, the

*maximal ideal* of

and

, the

*residue field* of

(a finite field). We define the

*normalized -adic absolute value* to be

. It is equivalent to absolute value we started with.

(Product formula)
For

,

for almost all places

and

Let

be a nonempty finite set of places of

containing the set

of archimedean places. The

*ring of -integer* is

. This is a Dedekind domain. For

, we denote

,

and

.

Every maximal ideal of

is of the form

for

.

For

,

. For any prime

,

.

Let

be a number field.

is called the

*ring of integers* of

. It is the integral closure of

in

.

Let

. Then taking the usual degree of rational function gives exactly the valuation associated to the closed point

. Moreover,

. A homomorphism

gives a map

. We know that

and

is the integral closure of

in

.

09/10/2012

Let

be a global field and

be a finite set of places of

containing

. Then

is a Dedekind domain. We denote the group of fractional ideals of

by

. Then

. Any

generates the principal fractional ideal

. Let

be the subgroup of principal ideals. We define the

*-class group* of

to be

.

##
Extensions of global fields

Let be a finite extension of global fields and be a finite set of places of containing . Let be the preimage of . Then natural map makes a finite projective -module of rank .

Fix a place , we factorize . Then the 's correspond to the places of restricting to .

Denote

. The number

is called the

*ramification index* of

, denoted by

. The number

is called the

*residue degree* of

. We have

.

is called

*unramified* at

if

for all

's,

*totally split* at

if

and

*inert* at

if

and

.

Now fix an archimedean place . Then is either 1 or 2.

is called

*unramified* if

and

*ramified* otherwise. Similarly we have

.

Suppose is Galois with , then acts on given by . This action is transitive on the fibers of .

The

*decomposition group* of

is defined to be the stabilizer of

in

. We have

. In particular, when

is abelian the

's for all

coincide, we simply denoted it by

.

When

is archimedean and

is ramified, we denote

, where

is the complex conjugation. These complex conjugations are related by

.

When

is non-archimedean,

acts on the residue extension

and gives a map

. This map is surjective and the kernel is called the

*inertia subgroup*, denoted by

. Similarly we have

. Counting shows that

. So

is unramified if and only if

, if and only if

.

Suppose

is unramified. The generator

of the cyclic group

is called the

*Frobenius*, denoted by

(you may think it as the analogue of the complex conjugation

. Similarly we have

. In particular, when

is abelian we have a unique Frobenius attached to

, denoted by

.

##
Valued fields

A

*valued field* is a pair

, where

is a field and

is a nontrivial absolute value on

. We endow

with the metric

. We say that

is

*complete* if

is complete under this metric.

A

*valued field extension* is a map

, i.e., a homomorphism

such that

.

(Gelfand-Mazur)
If

is a complete archimedean field. Then

is isomorphic to either

or

.

Let

be a non-archimedean valued field. The ring

is called the

*ring of integers*. It is a valuation ring with valuation group contained in

. Denote its fraction field by

. We similarly define the

*maximal ideal* and the

*residue field* .

We say

is

*discretely valued* if

is discrete (equivalently, equal to

for some

). Any element

such that

is called a

*uniformizer*. Let

be the unique valuation such that

.

The famous Hensel's lemma holds for any valued fields (but the proof in this generality is different from the discrete valued case).

(Hensel's lemma)
Suppose

is monic and

such that

and

. Then there exists

such that

and

.

09/12/2012

Let

be a complete discretely valued field and

be a complete discretely valued field extension of

. Let

be a uniformizer of

. We define

to be the

*ramification index* of

and

the

*residue degree*. We say that

is

*unramified* if

and

is separable (so

is separable), or equivalently,

is etale. We say that

is

*totally ramified* if

. Notice that

and

are multiplicative in towers

. So

is unramified (resp., totally ramified) if and only if

and

are unramified (resp. totally ramified).

We say

is

*tamely ramified* if

and

is separable, and is

*wildly ramified* otherwise. In particular, every unramified extension is tamely ramified.

##
Local fields

A

*local field* is a locally compact complete valued field

. So

and

are locally compact Hausdorff topological group.

Let

be a non-archimedean local field and

be a finite Galois extension with

.

acts on

by isometries, so

preserves

for any

and the action induces a map

. We define

to be the kernel of the map

, called

*higher ramification groups* (in the lower numbering). The higher ramification groups give a filtration

is called the

*inertia group*. We have

,

. In particular,

and

is totally ramified.

The fixed field of

is the maximal tamely ramified sub-extension

and

is the prime-to-

part of

.

is called the

*tame inertia group* and

is called the

*wild inertia group*.

is cyclic of order prime to

and

is the unique

-Sylow subgroup of

.

For

we define a function

, where

,

and

for

. Then

is a piecewise linear continuous increasing convex-up function. We define

to be the inverse function of

and define the

*higher ramification groups* (in the upper numbering)

. Then

and

, and

for

. For more details, see Chapter 4 of Serre's

*local fields*.

##
Topological groups

A continuous map

of topological space is called

*proper* if

is compact for any

compact,

*open* if

is open for any

open,

*closed* if

is closed for

any closed.

- A continuous map between locally compact Hausdorff spaces is proper if and only if it is closed with compact fiber.
- If and are proper and is Hausdorff, then is proper.
- If and are open, then is open.

09/14/2012

A *topological group* is a group with a topology under which the multiplication and the addition are continuous.

The following basic properties of topological groups are easy exercises.

For any subgroup

, we endow the coset space

the

*quotient topology*, i.e.,

is open if and only if

is open. By definition, if

is continuous homomorphism, then

is continuous.

- Let be a surjective continuous homomorphism. Then is a quotient map if and only if is open.
- Let be a closed continuous homomorphism of Hausdorff topological groups, then is open if and only if is open.
- Let be a surjective continuous homomorphism and . Suppose is a continuous homomorphism such that . Then is a homeomorphism and is a quotient map. In particular, when is abelian, we have an isomorphism of topological groups .

Let

be a locally compact Hausdorff topological group. A

*Haar measure* on

is a nonzero Radon measure

on

such that

for any

and any measurable subset

of

.

(Haar)
The Haar measure exists and is unique up to scalar multiplication.

The Lebesgue measure on

satisfies

and is a Haar measure. The standard measure on

is a Haar measure.

Let

be any local field and

be a Haar measure on

. For

, set

, then

is also a Haar measure. It follows from Haar's theorem that

, where

is a constant.

.

The archimedean case is obvious. Suppose

is non-archimedean. Since

is compact open, we know that

is a positive real number. Thus it is enough to show that

, Replacing

by

, we may assume that

. It easy to see (via the filtration

) that

. Hence

is a disjoint union of

cosets of

. Therefore

using the left-invariance of

.

¡õ
Let

be local field and

be a Haar measure. Then

is a Haar measure on

, i.e.,

is left-invariant.

##
Profinite groups

A

*profinite group* is a (filtered) inverse limit of finite groups

. We endow

with the product topology, which makes it a compact and Hausdorff topological group. Then

is a closed subspace, hence is also compact and Hausdorff and becomes a topological group under the subspace topology (equivalently, the weakest topology such that the projections

are continuous). Moreover, if

then there exists a projection

, hence

and

are disjoint open and closed subsets, we conclude that

is

*totally disconnected*.

In fact, we can characterize the profinite groups topologically as follows.

A topological group

is profinite if and only if it is compact and totally disconnected. In this case,

, where

runs over all open normal subgroups of

Let

be any group. Then the

*profinite completion* of

is defined to be

, where

runs over all normal subgroup of finite index of

.

is profinite and the natural map

has dense image. Any homomorphism of

, for

profinite, factors through

.

Let

be a field, then

is a profinite group. Hence any homomorphism

factors as

.

.

For any non-archimedean local field

,

and

are profinite.

If

are profinite, then the product

is also profinite.

09/17/2012

##
Infinite Galois theory

Let

be a field. We say

is

*separably closed* if there is no finite separable extension of

. We define the

*separable closure* of

to be an algebraic field extension of

which is separably closed. Any two separable closure are isomorphic.

The

*absolute Galois group* of

is defined to be

.

(infinite Galois theory)
Suppose

is Galois. Then there is a bijection between subextension

and

*closed* subgroups of

under the Krull topology given by

and

. Moreover,

is Galois if and only if

is normal. In this case, we have

as topological groups.

We say

is

*abelian* if it is Galois with abelian Galois group

. In this case, it is easy to see that any subextension of

is abelian.

Let

be a topological group. Then

is a normal topological subgroup. The

*topological abelianization* of

is defined to be the Hausdorff topological group

, i.e., the maximal Hausdorff abelian quotient of

.

Now let be a global field and be a (possibly infinite) Galois extension.

For any

, there exists a place

of

such that

and

acts transitively on such

.

The finite case is known. Assume that

is infinite. We have a

-algebra homomorphism

induced by

. There is a unique absolute value

on

extending

and restricting

to

gives a

. Now suppose

and

are two such places of

. Then

is nonempty. So the inverse limit with respect to finite Galois extensions

is again nonempty.

¡õ
The

*decomposition group* at

of

is defined to be the stabilizer

. This is a closed subgroup of

.

Suppose

is non-archimedean. Let

be the the residue field of

representing

. This is an algebraic extension of

. Then

acts on

and give a continuous homomorphism

. This map at each finite level is surjective, hence the image is dense. But

is compact, so this map is actually surjective. The kernel

of this map is called the

*inertia group*. It is a closed subgroup of

and is equal to

. We say

is

*unramified* in

if

for any

. In this case, by surjectivity, we have a

*Frobenius element* whose image is

. When

is abelian, this doesn't depend on

and we have a Frobenius element

.

Define

. Then

, where

runs over all finite separable extensions of

. Moreover,

is Galois and the composition

is an isomorphism of topological groups, with all maps being bijective.

Let

, there exists

such that

. By Krasner's lemma, for

,

. Set

. Then

. Now since

, where

runs over finite Galois subextensions, we know that any

is Galois, hence

is Galois. The composition map

is an inverse limit of isomorphisms, hence is a topological isomorphism. The injectivity of

follows from the fact that

is dense in

.

¡õ
09/19/2012

Suppose

. Then

.

By definition,

. Let

be a monic irreducible separable polynomial and

be a root of

. Let

with coefficients close to

and the same degree as

. Then

is small. On the other hand, write

, where

. Then

can be made as small as possible. Now Krasner's lemma implies that

. Comparison of degrees shows that

. So

is separable and irreducible over

, hence over

. We conclude that

, thus

.

¡õ
.

is always open.

is proper, hence is closed. It is enough it show that

is open is

by Exercise

3. Since we already know that the image is closed, it suffices to show that

is of finite index. This follows from the norm-index formula. In fact, homological methods can show that

(and equality holds if and only

is abelian).

¡õ
The unramified case is relatively simpler.

If

is unramified, then

. Hence

.

is clear. Since the image is closed in

, it suffices to show that the image is dense. Any

acts by isometry on

,

. So

, where the last equality holds because

is unramified. The norm map induces a map

, which is coincides with the norm map on the residue fields. By a counting argument we know that the norm map on finite fields is surjective. The norm map also induces a map

, which coincides with the trace map, so it is nonzero and

-linear (trace is always surjective for finite separable extension). This concludes that the norm map has dense image.

¡õ
##
Adeles

Let

be a family of topological spaces,

be open subset defined for almost all

. We define

and endow

with the topology given by the base of open subsets

The topological space

is called the

*restricted topological product* of

with respect to

. Notice that this topology is different from the subspace topology induced from the product topology.

The following lemma is easy to check.

Let

be a finite set of indices containing all

's such that

is not defined. Then

is open in

and the subspace topology on

is the product topology on

.

The following proposition partially explains the reason of introducing the restricted product.

If

's are locally compact Hausdorff and the

's are compact, then

is locally compact Hausdorff.

Notice that

is locally compact: it is a product of a locally compact Hausdorff space and a compact space. Then result then follows from the fact that

.

¡õ
Let

be a global field. The

*ring of ideles* is defined to be the restricted product of

with respect to

for

non-archimedean. It is a subring of

and is a locally compact Hausdorff topological ring.

Since

, we have

. The similar identification works for general number fields when replacing

by

. We also have

.

is a discrete closed subgroup and the quotient

is compact.

09/21/2012

We will omit the tedious measure-theoretic check of the following the lemma.

There exists a Haar measure

on

such that

where

is the normalized Haar measure on

such that

.

##
Ideles

The group of

*ideles* is defined to be

. The embeddings

defined by

gives an embedding

. The quotient

is the called the

*idele class group* of

.

The

*idelic norm* is defined to be the homomorphism

given by

. The

*unit ideles* is defined to be the subgroup

.

- is continuous. Hence is a closed subgroup.
- When is a number field, is surjective and open, hence is a quotient map.

When is a global function field, the image of can be described as follows.

Let

be a global function field with constant field

and

. Then

.

For

, then multiplication by

scales

by

.

It follows from the case of local fields (Lemma

1) and the Lemma

6.

¡õ
is a Haar measure on

.

From this we can obtain a slick proof of the product formula.

09/24/2012

(Theorem 2)
Consider the measure

in Remark

42. Suppose

, then

Since

is

*invariant* under multiplication by

, this integral is equal to

Hence

.

¡õ
Let

. Then the fractional ideal

where

determined up to sign. Hence

. But

implies that

. Replacing

by

if neccesary, there exists a unique

such that

as well. Hence

is a fundamental domain for

. Therefore

as topological groups. Hence

as topological groups. Now

also embeds into

, hence

as topological groups (this can also be seen directly by extracting all

-powers of

).

##
Adelic Minkowski's theorem

The classical Minkowski's theorem says that for a compact convex and symmetric around 0 region , implies that there exists a nonzero such that . The following is a reformulation.

(Minkowski)
Let

be a

-dimensional

-vector space and

be a lattice. Suppose

is a Haar measure on

constructed from the counting measure on

and the volume one measure on

. If

is compact convex and symmetric around 0, then

implies that

.

The idea of the proof of the following adelic version is essentially the same as the classical version.

(Adelic Minkowski)
Let

be a global field and

. Then

is compact. There exists

depending only on

such that if

, then

.

(Strong approximation)
Let

and

. Then the diagonal embedding

has dense image.

We claim that there exists

such that

. This follows from the fact that

is open (Exercise

3),

and

is compact.

For such a , let , and . Then for

- , we choose such that .
- and , we let .
- , we choose such that has , where is the constant in the Adelic Minkowski's Theorem 15.

By Adelic Minkowski's Theorem 15, there exists . Write such that and . So , then . Then , where and

- for any by construction.
- for and .
¡õ

09/26/2012

is closed in

and the subspace topology on

from

coincides with the subspace topology from

.

First we show that

is closed in

. For

, the products

will eventually be decreasing. So

is well-defined. Suppose

, then

. There are two cases:

- . Let be a finite set of places such that and for all . For and , we let and Then is an open neighborhood of in . For small enough, .
- . Let be a finite set of places such that
- for .
- for . This implies that if , and , then .
- .

For , let be a small open neighborhood of small enough such that for . Then is a neighborhood of in . Let . If for some , then . If for all , then . Hence . This concludes that is closed in .

Now is continuous and has closed image. So we need to show that any neighborhood of in contains for some a neighborhood of in . By homogeneity, we may assume . The basic open neighborhood of in is of the form , where and is an open neighborhood of 1 in . Shrinking the 's we may assume for any . We claim that works, i.e., . If for some , we have , then , thus a contradiction.
¡õ

Let

be a global field. Then

is a discrete closed subgroup and

is compact.

The assertion that

is discrete and closed follows form the case of

(Theorem

12) since the topology on

is finer than

. It remains to prove the compactness of

. By the previous lemma, if

is compact, then

is compact in

. So it suffices to show that there is a surjection

for some

compact. Let

be as in the Adelic Minkowski's Theorem

15 and choose

such that

. We claim that the compact set

works. Let

, then

. It follows from the Adelic Minkowski's Theorem

15 that there exists

, i.e.,

for any

. Now

, hence

. This concludes the surjectivity of

.

¡õ
##
Classical finiteness theorems

Let

be a finite set of places. We denote

an open subring of

. Similarly, we denote

an open subgroup of

. We have

and

.

Recall that the -class group (Definition 11) is , the fractional ideals of quotient by the principal ideals. We have a natural surjection with kernel and . Hence we have an isomorphism

09/28/2012

##
Idele class groups

Let be a finite extension of global fields. We know that as topological rings. In particular, is a closed embedding (i.e., a homeomorphism onto a closed subgroup). Hence is also a closed embedding.

The natural map

is a closed embedding.

For any

,

. Hence

. When

is a global function field, we have two exact sequences

Since

is compact, the first vertical map is a closed embedding. The last vertical map is also a closed embedding. Now the middle term is infinite union of these closed embeddings, hence is also a closed embedding. The number field case is similar:

The same argument shows that the middle map is a closed embedding.

¡õ
- For any , is a closed embedding.
- If is a finite set of places and . Then is
*not* a closed embedding.

Notice that is a finite -module, we obtain a norm map , compatible with the norm . Using , we know that for any , the norm map is also compatible with the local norm . Moreover, for any .

is continuous, open and proper.

It is continuous since each local norm is continuous and

(hence the inverse image of a basic open subset is open). For the properness, we use the splitting

and

. Then

is the norm on the compact factors

and identity on

or

, so it is a product of two proper maps, hence is proper. To prove the openness, we use the fact that local norms are open (Theorem

11) and the local norm is surjective for unramified local extensions (Lemma

4) to conclude that the image of a basic open subset is open.

¡õ
The map

is continuous and proper.

Our next goal is to describe the connected component of 1 in ideles class group (which turns out to be exactly the kernel of the global Artin map by class field theory).

First suppose is a number field. Then the connected component of 1 in is where and are the numbers of real and complex places of . It is divisible, i.e., is surjective for any . Let be its closure in . Then is a closed connected divisible subgroup (the divisibility follows from the fact that is proper, thus closed).

Suppose

is a number field. Then

is the connected component of 1 in

and

is profinite. Moreover,

is the set of all divisible elements in

.

Notice that

is finite and

, we know that the natural map

has finite cokernel. Let

be the image of this last map. Since

is profinite, we know that the image

is also a profinite group (Remark

30). Since

is of finite index, it is also open in

. Combining the fact that

is compact and totally disconnected, we find that

is also compact and totally disconnected, thus is profinite.

Let be the connected component of 1, then is killed under the map to by totally disconnectedness, therefore . But is already connected, this shows that . Every divisible element maps to 1 in since profinite group has no nontrivial divisible elements, so must lie in . But is already divisible, hence it consists of all divisible elements of .
¡õ

Now consider the global function field case.

Suppose

is a global function field. Then

is totally disconnected and has no nontrivial divisible elements.

Then

is an open neighborhood of 1 in

, hence

is an open neighborhood of 1 in

. Hence

is totally disconnected, thus

is profinite. Hence

has no divisible elements.

¡õ
##
Cyclotomic extensions

Let be any field and . Let be a primitive -th root of unity. Then , the splitting field of , is separable, hence is Galois. Let . Then any is determined by its action on , thus we obtain a injection This in particular shows that is abelian. The map is functorial, i.e., for any field extension , we have a commutative diagram

For

,

and

. So

forms a filtered directed system. We define the

*maximal cyclotomic extension* to be

.

10/01/2012

When

or

, the Kronecker-Weber theorem says that

. However, this is not true for general number fields. For example, for

, the extension

is abelian over

but is not even Galois over

, hence

cannot be contained in a cyclotomic extension of

since

is abelian over

for any

.

Suppose

, where

. Suppose

and

is the order of

. Since

, we know that

. Therefore

and

is a degree

extension of

. We have

,

and

is given by

.

Suppose

and

, then

and

sends the complex conjugation to

as

.

Suppose

is a non-archimedean local field. If

, then

is unramified. Indeed, if

is any finite extension, Hensel's lemma implies that

if and only if

. Hence

is the unramified extension with residue field

. We have

as the case of finite fields.

Suppose

is a global function field, then

. However, it is not clear whether this is the maximal abelian extension (indeed, not in general).

Let

be a number field and

. Then

is ramified at most over finite places

and ramified at the real places if

. For

,

, where

, due to the compatibility of

with respect to the inclusion

, and the decomposition group

maps to

. For

,

maps to

for

. The question remaining is what

takes

for

(which can be answered by Artin reciprocity law).

Now consider the case , we have . is ramified at if and only if (and is even for ). In fact, is totally ramified in since . Write , then is the compositum field of (where is unramified) and (where is totally ramified). By Chinese remainder theorem, we know that hence these two fields are linear disjoint. Comparing ramification index shows that the prime above of is totally ramified in and the prime above in is unramified in . So is the maximal unramified subextension at in . Hence maps isomorphic to under and given by . Therefore maps isomorphically to under .

Let

be a finite subextension. Then there exists a unique smallest

(the notation comes from German word

*Führer*) such that

. Moreover,

if and only if

is ramified in

.

Since

is finite, there exists an

such that

. Since

, the gcd of all such

's is the smallest

. If

, then

is unramified in

. Conversely, if

is unramified in

, we write

, then the restriction map on inertia groups is

. Hence

is contained in the fixed field of

of

, i.e.,

.

¡õ
##
Artin maps

Recall the following commutative diagram We know that for , maps surjectively to the inertia group and maps surjectively to the wild inertia subgroup, i.e., the -Sylow subgroup of . For , the element (with 1 at the place ) maps to , which is equal to .

Let

be a finite abelian extension

. Define the

*Artin map* by sending

to

and

as the restriction

. It is a continuous surjection. It follows that

maps surjectively to the inertia group

and

maps to

.

10/03/2012

,

. When

is unramified is

,

(opposite to the usual Artin map), where

is the image of

.

We already know that

. Without loss of generality we may assume

. Suppose

is unramified, then

maps to

, i.e.,

. For arbitrary

, we write

. Then

. Because any prime

is unramified in

, we also know that

. Hence

We know that

This completes the proof.

¡õ
Now for any

with infinite degree,

makes sense by taking the inverse limit over

finite over

.

is obviously continuous by construction. Since

is compact and the image is dense (surjective on every finite

, we find the

*Artin map* is surjective.

The following proposition summarizes easy properties of the Artin map . We will see how they generalize for any global field.

- is a bijection between the finite subextension of and open subgroups (of finite index) of . Indeed, any open subgroup contains some , hence is a subgroup of , hence corresponds to a finite extension .
- is continuous (and surjective onto ).
- , the connected component of .

Now let us turn to the local case .

Suppose , then is unramified over . Recall that sending to , so , where is the order of in . On the other hand, is totally ramified of degree . Hence is an isomorphism. Hence for general , we have and the inertia subgroup .

Taking the inverse limit of the exact sequence we obtain two isomorphic exact sequences

We define the

*Artin map* sending

This is a continuous map with dense image and maps

surjective onto the inertia group.

10/05/2012

The following important proposition is left as an exercise.

(Local-global compatibility)
We have the following commutative diagram

##
Weil groups

Let be a nonarchimedean local field (resp. a global function field) and be the residue field (resp. the constant field). The residue field (resp. constant field) of is . Sow we have a continuous surjection . Denote its kernel by (which is the inertia group in the local case). As a group, the *Weil group* is simply , i.e., the elements in which induces integral powers of Frobenius on . However, as we have seen in Remark 53, may not be open in under the subspace topology from . But there exists a finer topology on such that is open in . under this finer topology, (in the local case) or (in the global function field case) will be isomorphisms of topological groups.

More precisely, suppose we have a short exact sequence of profinite groups and .

There exists a unique topology on

such that

is open in

,

has the subspace topology under

and any splitting of

induces an isomorphism

of topological groups.

See the handouts for details.
¡õ

Under this topology on the Weil group , is continuous and has dense image (but is not a homeomorphism onto its image). Moreover, the topology is compatible under abelianization as in following proposition.

Let

be the abelianization of

and

:

Then

with

the abelianization as topological groups (but not for

).

The following theorem allows us to use Weil groups as a replacement of to classify finite abelian extensions of local or global function fields.

The maps

and

gives a bijection between open subgroups of

and open finite index subgroup of

. For any

an open subgroup,

is isomorphic to

as discrete coset spaces.

This bijection extends to a bijection between closed subgroups

of

such that

or

has finite index in

, and closed subgroups of

. This will allows us to partially classify infinite abelian extensions of local fields and global function fields.

##
Statement of global class field theory

Property (a) in Theorem

21 uniquely characterizes

.

Observe that if

is a finite set of places of

. Then

is dense in

by the weak approximation (Remark

47). If

is a finite abelian extension, we let

contain

and all the ramified places. Then

is determined by (a), hence by continuity

is determined by (a).

¡õ
Assuming Theorem 21, let us prove the following "real version" of the existence theorem (and please hope for the "real real version").

(Existence Theorem)
is an inclusion reverse bijection between finite abelian extensions

and open subgroups of finite index in

.

It suffices to show the injectivity. Suppose

and

are finite abelian extensions such that

. Let

and

be the corresponding open subgroup in

, then

. Hence

. Suppose

, then there is an neighborhood

since

and

are both open and closed. But

is dense, this would contradict

. We conclude that

, thus

.

¡õ
10/10/2012

Suppose is a number field and be the connected component of . We have shown that is profinite (Lemma 10). is profinite, thus contains no divisible elements, hence . But the image of is dense, we know that is *surjective* in this case. On the other hand, is profinite implies that the intersection of all open subgroups is trivial. We find that the intersection of all open subgroups (of finite index) of is , as any subgroup of finite index contains divisible elements . Therefore by the existence theorem. Namely, we have shown

Suppose

is a number field, then

induces an isomorphism of topological groups

.

We thus deduce a stronger version of existence theorem for number fields.

The map

is a bijection between

*all* abelian extensions

and

*closed* subgroups of

such that

. Under this bijection,

is called the

*class field* of

.

Now suppose is a global function field. We have an isomorphism of topological groups , a product of a discrete group and a profinite group. Then the intersection of all open subgroup is trivial, hence is *injective*.

But is not surjective, indeed we claim that if , then , an *integral* power of the Frobenius. In fact, for a finite extension , then is finite abelian and unramified everywhere and (Theorem 13, or the base change of an etale map is etale). It suffices to show that . For , lifts , By definition, as is unramified everywhere and . In other words, we have proved the following diagram commutes

It follows that . We claim that is actually an isomorphism. We already know it is injective, so it suffices to show the surjectivity. Choose such that . Then there exists a unique continuous homomorphism sending 1 to by the universal property of profinite groups. Hence as topological groups. Under this identification, is simply . Let . Then is a closed subgroup and , hence the closure of in is . Since the image of is dense, it follows that , which proves the surjectivity.

We have proved

induces vertical isomorphisms of topological groups

So

can viewed as the Weil group

and

is the profinite completion of

.

gives a bijection between

*all* abelian extensions

such that the constant field extension

is either finite or equal to

, and

*closed subgroups* of

.

10/12/2012

##
Norm and Verlagerung functoriality

Let be a global field and be any finite separable extension. gives us a *canonical* map . We also have the compatibility of local and global norms:

(Norm functoriality)
The follows diagram commutes:

Suppose

is finite abelian, then

. In other words,

.

Take

in the previous theorem.

¡õ
(Global norm index inequality)
Suppose

is a global field and

is a finite and separable extension. Then

.

We will give an easy analytic proof later (cf. Exercise

17). In fact, the cohomological proof will even give a division relation.

¡õ
(Existence theorem III)
Suppose

is a global field and

is a finite and separable extension. Then

if and only if

is abelian, in which case

.

It remains to prove the "only if" direction, which is left as an exercise (hint: the left-hand-side is always

.

¡õ
Now let us briefly turn to the verlagerung functoriality of the Artin map.

Let

be a group and

be a subgroup of finite index. For any

, let

be the smallest

such that

. This only depends the choice of

in

, a finite double coset. Write

be the coset representatives. We define the

*verlagerung* (or

*transfer*)

. Then

is a group homomorphism. In terms of group cohomology, this is the restriction map

and functorial in

.

Suppose is a field and is finite separable. Then is of finite index (depending on the choice of an isomorphisms ). We then obtain the verlagerung , a continuous map of the topological abelianization of and , which does not depend on the choice of .

Suppose

is a global field and

is a finite separable extension. The following diagram commutes:

Suppose

is finite Galois and

. It suffices to show at the finite level. The remaining check will be an easy calculation which we leave as an exercise.

¡õ
##
Statement of local class field theory

10/15/2012

Analogously, the local existence theorem will follow from the local norm index inequality.

(Local norm index inequality)
Suppose

is a local field and

is a finite and separable extension. Then

.

(Existence Theorem)
Suppose

is a local field and

is a finite and separable extension. Then

if and only if

is abelian, in which case,

.

The local-global compatibility will follow from defining the global Artin map via "gluing" local Artin maps.

(Local-global compatibility)
Suppose

is a global field and

. Then we have a commutative diagram

If

is finite abelian and

, then

is also finite abelian and we have a corresponding commutative diagram at finite level.

We can then derive part of (a) in global class field theory (Theorem 21) using the local-global compatibility at archimedean places.

Suppose

is a number field and

is a real place. If

is finite abelian, then

is unramified in

if and only if

kills all

. Otherwise

kills

and

is the complex conjugation.

##
Ray class fields and conductors

Now let us discuss the classical formulation of class field theory in terms of ideal classes.

Let

be a global field. A

*modulus* of

is a formal product

, where

and

for almost all

; for

a real place,

;

a complex place,

. We denote by

the

-component of

. We write

or

if

. The modulus is used to keep track of the ramification of the places of

in some sense.

Let

, we say

if and only if

for any

. This is a multiplicative condition, so it makes sense to say that

whenever

.

For

, each modulus is of the form

or

, where

. It follows easily that an idele

if and only the finite part

(which can be thought of the usual congruence relation

), and

if we further have

.

Suppose

is a modulus, we define

, namely

It is an open subgroup of

and

if and only if

. We know that

forms a cofinal filtered system of open subgroups in

. This generalizes Proposition

4.

For

and

, we have

. We have a surjection

and the kernel is exactly

. Therefore

. Similarly, when

, we find that

.

10/17/2012

There are isomorphisms

We define

(it properly contains

) and

. Then we have an injection

It is actually an isomorphism by weak approximation. Now define the homomorphism

which is obviously a surjection with kernel

. Taking quotient by the image of

, we obtain that

.

¡õ
The

*ray class field* of the modulus

is the class field

corresponding to

. When

is a number field,

is finite abelian. When

is a global function field,

contains

since the image of

has trivial image when projected to

.

When

, we have

and

and the following diagram commutes

We summarize the easy properties of ray class fields as follows.

- is unramified at all .
- If , then .

Suppose

is a finite abelian extension. We say a modulus

of

is

*admissible* for

if

. The gcd of all admissible modulus of

is called the

*conductor* of

. It follows that

.

When

.

is essentially the same as Proposition

3, except that

if and only if

is totally real.

is admissible for

if and only if

.

The "if" direction follows from the fact that

. For the "only if" direction, suppose

, then the local-global compatibility implies that

for

nonarchimedean (and a similar thing for archimedean places), hence by the definition of idelic norm,

. Therefore

.

¡õ
is ramified in

if and only if

.

Let

, where

. If

is ramified, the

, hence

, hence

, which implies that

. Now suppose

is unramified, then for any

,

. Then we can find a modulus

such that

and

. It follows that

by the previous lemma and the definition of

.

¡õ
##
Ideal-theoretic formulation of global class field theory

Let

be a finite abelian extension and

be a modulus which is divisible by all ramified places. We define the

*Artin symbol*
is admissible if and only if

. (This is the way Artin originally introduced the notion of admissible moduli. The existence of admissible moduli is truly surprising and is the key difficulty of class field theory!)

10/19/2012

Suppose

is a finite extension and

is a modulus of

. We denote by

the free abelian group generated by prime ideals of

whose restriction to

are coprime to

. The usual ideal norm restricts to a group homomorphism

. We denote the image of

by

.

Let

be a finite extension. Suppose

is finite abelian and

. Then we have a commutative diagram

where

is a modulus of

divisible by the primes of ramified in

,

is a modulus of

divisible by primes ramified in

or restrict to primes of

ramified in

.

##
Weber L-functions

Before stepping into the cohomological proof of class field theory, we will discuss various applications of class field theory in the following several weeks. From now on we will assume is a number field for simplicity (though some results are also valid for function fields). Write the degree .

We defined the

*Dedekind zeta function* where

is the absolute norm.

We omit the proof of the simple analytic property.

is analytic on

except a simple pole at

.

Let

be a modulus of

and

be an ideal class. We define

A similar analytic property holds for too.

is analytic on

except a simple pole at

. The residue at

depends on the modulus

but not on ideal class

.

Recall that for a finite *abelian* group, we have the notion of *Pontryakin dual* consisting of characters of , and the following elementary properties holds:

- canonically.
- If , then
- If , then

The

*Weber -function* for a character

is defined to be

It follows from the previous theorem that

is analytic on

except a possible simple pole at

. When

, it is actually analytic at

by the previous proposition, since all the residues of

at

are the same.

Consider

and

. Then

is simply the

*Riemann zeta function*. A character

is the same thing as a

*Dirichlet character* . Then

is simply a

*Dirichlet -function*, where we extend

on

by letting

whenever

.

Now class field theory easily imply the following result on special values of Weber -functions.

if

.

By global class field theory, there exists a class field

such that

. So

can be viewed as a character of

. Let

be a modulus of

divisible exactly by the primes of

restricting to

. Then as a special case of the lemma below (

)

The result then follows since each of the

-functions

and

has a simple pole at

.

¡õ
10/22/2012

Let

be a finite abelian extension. Suppose

is an admissible modulus for

and

is exactly divisible by primes of

restricting to primes dividing

. Then

where

runs over all characters

.

This equality actually holds at the level of local Euler factors. Say

, then

is unramified and

, we claim that

Write

, then

. The right hand side becomes

Write

for short. Taking the logarithms of both sides, it reduces to show that

Notice that

and

has order

in

. Then the character

of the Pontryagin dual of

is nontrivial if and only if

. Therefore the right-hand-side is simply

which coincides with the left-hand-side.

¡õ
Consider a one-dimensional Galois representation

. Then

factors through a finite cyclic extension

, where

. Let

be the conductor of

, then

induces a character

. By definition,

since

if and only if

is unramified in

. In other words, Weber

-functions can be viewed as the same thing as one-dimensional Artin

-functions via

*class field theory*. Notice Weber

-functions involves geometry of numbers (ideal classes, etc.), which are crucial for establishing the analytic properties. On the other hand, the functoriality properties of Artin

-functions give handy ways to establish non-vanishing results of special values. This picture motivates the Langlands program of the study of general Artin

-functions via relating Galois representations and automorphic representations and class field theory can be viewed as the Langlands program for

.

Suppose

is a finite abelian extension with

. Then

(notice the left-hand-side is the regular representation of

). Thus we have

This is analogous to the previous lemma except that more local Euler factors are involved here.

General Artin -functions can be reduced to one-dimensional Artin -functions via Brauer's induction theorem.

(Brauer)
Suppose

is a finite group and

. Then there exists

,

subgroups of

and characters

such that

as virtual representations.

With the same notation,

##
Chebotarev's density theorem

Let

be a number field of degree

. Let

be a set of finite primes of

. We define the

*natural density* of

to be the limit (if it exists)

We define the

*Dirichlet density* to be

10/24/2012

The following basic properties of the (Dirichlet) density follows easily from definition.

- If has a density, then .
- If is finite, then .
- If is the disjoint union of and and two of , , have density, then so does the third one and .
- If and both have density, then .
- If has a density and ,then .
- If has density and is the complement of , then has density and .

Suppose

is a number field. Denote the

*degree* of a prime

by

. Prove that

has density 1. (Not to be confused with the density of of split primes of

in

, which will be shown to be

).

Let

be a modulus and

. Then

.

Notice that

On the other hand, since

is analytic at 1 whenever

, we know that

Expanding the last sum gives

The desired result then follows.

¡õ
When

,

, we have

and an ideal class is given by simply given by a residue class modulo

. From the previous proposition we recover the classical theorem of Dirichlet on primes in progressions:

. Namely, the primes are equidistributed modulo

.

(Chebotarev's density)
Suppose

is a Galois (but not necessarily abelian) extension of global fields with

and

. Let

and

be the conjugacy class of

in

(with

). Then

has density

. In other words, the Frobenius conjugacy classes are equidistributed in

.

Let us first show the case that

is abelian. Let

be the conductor of

. Let

be the kernel of

. Then

and

, where

is any preimage of

. Since

contains exactly

ideal classes, we know that

as needed. This is the only step where the usage of class field theory is crucial.

10/26/2012

For the general case, we let be the fixed field of under the cyclic group generated by . Then is a cyclic extension with Galois group and we can reduce the previous case as follows. Let be the primes of unramified in and . By the abelian case, we know that . Also, let be the primes in such that , then by Exercise 15. We claim that for any , . Assuming this claim, we know that as desired.

It remains to show the claim. Let be the set of primes of such that is unramified in and . For , write and . Since acts trivially on , it also acts trivially on . Therefore , i.e. . Since , we know that is the unique prime of over . On the other hand, given , let and be a prime of over . Since has order , we know that . Hence is the unique prime of over and . Thus . In this way we have exhibited a bijection between and .

Now let and a prime of over . We can choose . Then is the orbit of under the centralizer of . So which proves the claim.
¡õ

##
Split primes

Let

be a global field and

be a finite separable extension. We define

For

, we also define

.

Suppose

is

*Galois* and

. Then

and

have density

.

Use Chebotarev's Density Theorem

34 for

.

¡õ
- Suppose are finite separable. Then .
- Suppose is finite separable and is its Galois closure. Then . In particular, if and only if is Galois.

If

has density 1, then

.

Apply the previous exercise to the Galois closure of

.

¡õ
We can now prove the following results without using class field theory.

- for .
- (Global norm index inequality) for a finite extension of number fields.

Suppose

are finite

*Galois*,

. Then the following are equivalent:

- .
- .
- .

(a) implies (c) and (c) implies (b) are obvious. For (b) implies (a), notice that

. Therefore

and

.

¡õ
Suppose

and

are finite Galois,

. Then

if and only if

. In other words, a Galois extension is determined by the set of split primes.

##
Hilbert class fields

Let

be a number field. The

*Hilbert class field* of

is defined to be the ray class field

for

. Then

and

is unramified at every place (including archimedean places). The

*narrow Hilbert class field* of

is defined to be the ray class field

, where

is the product of all real places of

. Then

(the

*narrow class group*) and

is unramified at all finite places. Notice that

and

surjectis onto

.

(resp.

) is the maximal abelian extension of

which is unramified everywhere (resp. at all finite places).

If

is unramified everywhere, then

by Proposition

12. Hence

. Similarly for

.

¡õ
10/31/2012

Suppose

is finite extension of number fields and is totally ramified at some

. Then

.

Since

is abelian and unramified everywhere, we know that

. So it suffices to show that

and

are linearly disjoint over

(i.e.

. Suppose

with

the minimal polynomial of

. Let

be a monic polynomial dividing

over

. Since

is Galois,

splits into linear factors over

, which implies

also splits over

. Hence

, which is equal to

by the assumption that

is totally ramified at some place. Hence

and

is irreducible over

as needed.

¡õ
Consider

,

. We know that

using the previous proposition. When

is a prime, the famous criterion of Kummer asserts that

divides

(indeed equivalent to that

divides

) if and only if

divides the numerator of some Bernoulli number

. Such a prime

is called

*irregular*.

##
Artin's principal ideal theorem

(Principal Ideal Theorem)
Let

be a number field and

be its Hilbert class field. Then any ideal

becomes principal in

.

Using class field theory, it will reduce to the following purely group-theoretic theorem (we omit the proof).

(Furtwangler)
Let

be a finite group and

. Then

is trivial.

(Proof of the Principal Ideal Theorem)
Let

be the Hilbert class field of

. Then

is Galois since

is intrinsic to

. Notice that

is the maximal abelian subextension of

. Therefore

and

, where

. We have the following commutative diagram

By the previous theorem, we know that the

, hence

is principal.

¡õ
##
Class field towers

The principal ideal theorem motivates the following construction. Let be a number field and be the Hilbert class field of . The we obtain the *Golod-Shafarevich tower* Does stabilize (i.e., )? The answer in general is no.

The analysis on this tower may be easier if we consider a single prime at one time. Let be the maximal unramified abelian -extension of . We obtain a tower We now state a theorem of Golod-Shafarevich (for the proofs, cf. Cassels-Frolich Ch. IX).

(Golod-Shafarevich)
Let

be a number field of degree

. If

, then

where

is the

-rank of a finite group of

.

On the other hand, Brumer's theorem provides a lower bound on .

(Brumer)
Suppose

is Galois of degree

. Let

be the number of primes

such that

for any

above

. Then

Suppose

is Galois of degree

. If

, then

.

When

, it follows from the previous corollary that a quadratic field has infinite Golod-Shafarevich tower whenever the number of ramified primes is at least 8, e.g.,

where

has at least 8 different prime factors. In particular, there are infinite such quadratic fields.

##
Hilbert class fields of global function fields

Now suppose is a global function field. Then is the maximal abelian unramified extension of . In particular, and is infinite. To get better situation, we may ask what is the maximal unramified extension of with constant field . As class fields with constant fields corresponds to subgroups that surjects onto under , we know that a maximal unramified extension with constant field corresponds to a minimal subgroup such that there exists an element with . However, there may be more than one such minimal subgroup. The set of such subgroups forms a principal homogeneous space under (in geometrical terms, it is simply ). So there are such groups and hence there are maximal abelian unramified extension of with constant field .

11/02/2012

Nevertheless, the following construction gives a maximal unramified extension which is canonical in some sense. Choose with . Then modulo is independent on the choice of and the subgroup is canonically defined. Let be the class field of . We then have an exact sequence So is everywhere unramified of degree with constant field the degree extension of .

We end the discussion by applying a similar idea to prove a useful proposition concerning -adic characters of the Weil group of a global function field.

Let

be a Hausdorff group. A continuous character

is called

*unramified* if the class field corresponding to

is unramified at

.

Suppose

is a global function field of characteristic

and

is a finite extension of

for some

. Let

be a continuous homomorphism unramified outside a finite set

. Then there exists

and

continuous of finite order such that

.

Since

is abelian,

factors through

. Pick

such that

. Let

and

. Notice that

by construction. But

, it suffices to show that

is finite as

is finite. But

and by assumption

for

, it suffices to show that

is finite. But

has a finite index pro-

group and

has a finite index pro-

group. Because there is no nontrivial map from a pro-

group to a pro-

group, the image must be finite.

¡õ
##
Grunwald-Wang theorem

Let

be a global field. Suppose

and

contains the

-th roots of unity. If

is an

-th power in

for almost all

, then

itself is an

-th power in

.

Let

be an

-th root of

and

be its minimal polynomial. So

. If

is an

-th power in

, then

splits completely in

as

contains the

-th roots of unity. Since

is separable, it follows that

splits completely in

. Now the split primes

has density 1, hence

.

¡õ
- 16 is an 8-th power in and in for , but not in (hence not in , which is obvious).
- Let , then 16 is an 8-th power in for any place but not an 8-th power in .

(Grunwald-Wang)
Let

be a global field. If

is a

-th power in

for almost all

, then

is an

-th power in

, except potentially if

is a number field and

is not cyclic, where

with

odd.

The proof only uses the result on split primes (Corollary 10), so it is not really an application of class field theory.

11/05/2012

It suffices to treat the case that

is a power of a prime. In fact, suppose

with

coprime, then by the Euclidean algorithm we can write

. Suppose

, then

. We can further assume that

. If

and

such that

. Then

is purely inseparable over

. But

is separable for any

, which implies that

and

.

First consider the case that is cyclic of -power order. We know that splits into linear factors over by the previous proposition. Now look at the factorization over and choose a root of in for each . If for some , then for some and splits in (notice that is always abelian). Since is a cyclic of -power order, its subfields are totally ordered. Hence there exists an such that for any . Therefore splits in for almost all . By Corollary 10, we know that and has a linear factor over , i.e., .

The case and the follows since is always cyclic of 2-power order, due to the assumption in the number field case and the fact that every finite extension of the constant field is cyclic extension in the global function field case. It remains to prove the case odd and . Notice that the extension is always cyclic of -power order and we can apply the first case to find such that . Let , then is an -th power, where is the degree of . Because is coprime to , we know that itself is an -th power.
¡õ

Let us analyze the exceptional case in more detail. It is remarkable that we can write down the exceptional cases completely. In general when the local-global principle fails, it is quite rare that the failure can be completely classified.

Suppose is not cyclic. We choose a -th root of unity such that , and . Write .

- .
- .
- is cyclic of 2-power order.

Corresponding to the decomposition of

, we have the decomposition of

. The lemma follows immediately from this decomposition.

¡õ
Let be the unique integer such that and .

is cyclic for any

if and only if

.

If

is cyclic, then it contains a unique quadratic extension

as

. We find that

, hence

. If

, then

is cyclic for all

by the previous lemma. For

,

is also cyclic because

.

¡õ
Let

be a finite set. We define

.

The following key lemma characterizes all the counter examples.

is always cyclic, so by the Grunwald-Wang theorem there exists

such that

. Let

be the complex conjugation. We compute

So

is a

-th root of unity. But

, hence

is a

-root of unity. So we can write

for some

. Define an element

for some

to be determined. Then

If

is even, then we choose

so that

. Then

and

(as

), which contradicts our assumption. If

is odd, we choose

so that

. Replacing

by

, we may assume that

. Notice that

, we compute

If

, then

, hence

. So

, where

Notice that

. Hence

. Also

, hence the lemma follows.

¡õ
11/07/2012

We can actually show that . If not, then . Thus for some -th root of unity . But , which contradicts our assumption. Therefore if then is the disjoint union of and .

We have shown that in order to ensure to be non-cyclic, we necessarily have , are non-squares in . We shall show this obstruction always fails with the further condition (d) as in the following strong version of Grunwald-Wang theorem.

Suppose

is a number field and

, then (a)-(c) holds and

. We have to show that (d) holds, i.e., for

, then

. If

. Then we deduce that

by Euclidean algorithm. Hence there exits

and

. If

, then

, we know

. Otherwise,

, hence

.

For the other direction, we must show that if (a)-(d) hold, then . It is equivalent to showing that for any . Let . Condition (b) implies that is a degree four extension with Galois group . Moreover, is *unramified* away from 2 by construction. If is a palce of above . Then is quadratic. Consequently one of is a square in . If and , the one of is a square in by the assumption (d). In either case,

- if or is a square in . Then . Therefore .
- if , then , hence .
We conclude that .
¡õ

11/30/2012

##
Outline of the proof

###
Group cohomology

Suppose is any group. From an exact sequence of -modules, taking -invariants gives an exact sequence But the last map is not necessarily surjective, e.g., consider and the Kummer sequence The *group cohomology* functor , extends the above left exact sequence to the expected long exact sequence.

Dually, we can also take the -coinvariants (the largest quotient on which acts trivially) which does not preserve injectivity and the *group homology functor* fills in the corresponding long exact sequences.

When is a finite group, Tate defines the *norm map* . It maps into and factors through . Then we can connect both long exact sequence in cohomology and homology as Write , and . Then the five lemma gives a long exact sequence in Tate group cohomology in both directions.

When is cyclic, we further have , i.e., the Tate cohomology groups are periodic of period 2.

Let

be finite. Then the Hilbert 90 theorem asserts that

. In particular, when

is cyclic,

, which implies that

if and only if

(the classical formulation of Hilbert 90).

.

and dually

, where

is endowed with the trivial

-action.

###
Class formations

Class formations tell you all the group cohomology input in order to derive all the statements in class field theory (e.g., Artin maps).

Let be a local or global field. Suppose and is a -module. We say is a *continuous* -module if (e.g., and . Fix such an , for any Galois extension , we define and .

Suppose is Galois and , we have an *inflation* map given by the precompostion by the natural surjection on cocycles.

Suppose , we have a *restriction* map given by the precompostion by the natural inclusion on cocycles.

Using pure group cohomology, we will prove the following main theorem.

Cup product with

gives isomorphisms

. In particular, when

and

is abelian, we obtain that

. We define the

*Artin map* to be the inverse of this isomorphism.

###
Local class field theory

The main reference will be Serre's *local fields* and *Galois cohomology*. We set , then . The first axiom in class formation is simply Hilbert 90 and the second axiom amounts to proving that . Let (for local fields , but this als0 works for general complete discretely valued fields with quasi-finite residue fields, however, the existence does not work, the local compactness is still needed). For any field , we write .

We have an exact sequence

where

is the residue field of

.

The only non-formal part is the following

(in other words, any central simple algebra over

is split after an unramified base change.

When

is finite,

(in other words, there are no noncommutative central division algebra over a finite field).

When

is local,

.

The class formation machinery then proves local class field theory except the existence theorem. Proving the latter boils down to constructing enough abelian extensions using cyclotomic, Kummer and Artin-Schrier extensions (together with pure topological arguments).

###
Global class field theory

The main reference will be Cassels-Frolich and Artin-Tate. In global class field theory, the idele class group plays the role of in local class field theory. The first axiom in class formation is again Hilbert 90. For the second axiom, we will prove is the the product of local Artin maps and it kills all the global elements. By the exact sequence , the Brauer group of a global field fits into As we will see, the proof of the existence theorem (norm index inequality) interplays with the proof of Artin maps (class formation): first proved for cyclotomic extensions and then in general.