数学776讲义：1997年冬季版J.S. Milne的类域理论

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"J.S. Milne的《Class Field Theory》(1997版)是他在1997年冬季在密歇根大学数学776课程中的讲义，经过修订和扩展，基于1993年的早期版本（v2.01）。这份资料主要涉及数学、代数和数论领域。" 《Class Field Theory》是数学领域的一份重要文献，由J.S. Milne撰写，专注于类域理论这一深奥的主题。类域理论是数论的一个分支，它研究的是整数环上的理想类群与数域的伽罗瓦群之间的关系。在这些讲义中，Milne采用了一种折衷的方法来介绍这个理论，即不仅使用分析方法，也涉及同调理论。讲义共分为十二章，其中第二章介绍了群的同调理论，这是后续章节的基础。然而，第四、六和八章并不是阅读第三、五和七章的必要条件。对于不感兴趣于具体局部类域理论的读者来说，除了第一章的第一节，其他部分可以跳过。 Milne在书中没有回避使用分析或同调理论来阐述关键定理，即使这会使得证明更加复杂。他指出，有些主要定理，例如关于素数密度的定理或者与数域相关的同调群的定理，如果不借助这些工具，甚至无法表述。他选择在适当的时候提供多种证明方法，以增加对结果的理解深度。书中的参考文献标记为"Math xxx"的形式，表明这些是可以在指定网站上找到的其他课程笔记，提供了进一步学习和探索的资源。类域理论是理解数论中一些核心问题的关键，比如阿贝尔扩张、黎曼猜想的局部版本以及理想类群的性质。通过这些讲义，读者将有机会深入到数论的核心，了解如何利用抽象代数和拓扑概念来揭示数的奥秘。Milne的工作为学者和学生提供了一个全面而深入的视角，帮助他们理解这个领域的复杂性和美丽。

8CONTENTS

CHAPTER I

Local Class Field Theory:

Statemen ts of the Main Theorems and Lubin-Tate

Extensions

Local class ﬁeld theory classiﬁes the abelian extensions of a local ﬁeld. From a diﬀerent

perspective, it describes the local components of the global Artin map.

By a local ﬁeld, I mean a ﬁeld K that is locally compact with respect to a nontrivial

valuation. Thus it is

(a) a ﬁnite extension of Q

for some p;

(b) a ﬁnite extension of the ﬁeld of Laurent series F

((T )) over the ﬁeld with p

elements; or

When K is nonarchimedean, the ring of integers (alias, valuation ring) in K is denoted

by O

(or A), its maximal ideal by m

(or just m), and its group of units by O

. A generator of m is called a prime element of K (rather than the more customary

local uniformizing parameter). If π is a prime element of K, then every element of

can be written uniquely in the form a = uπ

with u ∈O

and m

=ord

(a).

The residue ﬁeld k of K has q elements, and its characteristic is p. The normalized

valuation on K is deﬁned by |a| = q

−ord

(a)

We ﬁx a separable algebraic closure K

of K. Throughout the chapter “extension

of K” will mean “subﬁeld of K

containing K”. Both ord

and |·|have unique

extensions to K

1. Statements of the Main Theorems

The composite of two ﬁnite abelian extensions of K is again a ﬁnite abelian exten-

sion of K. Therefore the union K

of all ﬁnite abelian extensions of K (in K

)isan

inﬁnite abelian extension whose Galois group is the quotient of Gal(K

/K)bythe

closure of its commutator subgroup. (See the appendix to this chapter for a review

of the Galois theory of inﬁnite extensions.)

Let L be a ﬁnite unramiﬁed extension of K.ThenL is Galois over K,andthere

is a unique element σ ∈ Gal(L/K) such that σα ≡ α

for all α ∈O

, i.e., such that

σ induces the Frobenius automorphism on the residue ﬁeld of L.Thisσ is called the

Frobenius element of Gal(L/K), and is denoted Frob

L/K

. It generates Gal(L/K).

(See Math 676, 7.41 et seqq.)

10 I. LOCAL CLASS FIELD THEORY

Theorem 1.1 (Local Reciprocity Law). For any nonarchimedian local ﬁeld,

there is a unique homomorphism

: K

→ Gal(K

/K)

with the following properties:

(a) for any prime element π of K and any ﬁnite unramiﬁed extension L of K,

(π)|L =Frob

L/K

;

(b) for any ﬁnite abelian extension L of K, Nm

L/K

) is contained in the kernel

of a → φ

(a)|L,andφ

induces an isomorphism

L/K

: K

/ Nm

L/K

) → Gal(L/K).

Denote Nm

L/K

)byNm(L

). Statement (b) says that, for every ﬁnite abelian

extension L of K, there is a commutative diagram

−−−→ Gal(K

/K)









τ →τ |L

/ Nm(L

)

L/K

−−−→ Gal(L/K)

with φ

L/K

an isomorphism. I shall refer to φ

(and φ

L/K

)asthelocal Artin map.

Other names in use: (local) reciprocity map, norm residue map (symbol), etc..

Theorem 1.2 (Local Existence Theorem). AsubgroupN of K

is of the

form Nm

L/K

) for some ﬁnite abelian extension L of K ifandonlyifitisofﬁnite

index and open.

The proofs of these theorems will occupy most of the rest of this chapter and of

chapter 3. First, we note an immediate consequence.

Corollary 1.3. The map L → Nm(L

) is a bijection from the set of ﬁnite abelian

extensions of K to the set of open subgroups of ﬁnite index in K

. Moreover

⊂ L

⇐⇒ Nm(L

) ⊃ Nm(L

);

Nm(L

· L

)=Nm(L

) ∩ Nm(L

);

Nm(L

∩L

)=Nm(L

) · Nm(L

Proof. Let

K be a ﬁnite abelian extension of K. According to the Theorem 1.1,

K/K

identiﬁes K

/ Nm(

)withGal(

K/K) in such a way that a subﬁeld L of

K is the ﬁxed ﬁeld of the subgroup Nm(L

)/ Nm(

)ofK

/ Nm(

). Moreover,

Theorem 1.2 shows that every subgroup H of K

containing Nm(

)isanorm

group. Therefore, so far as it concerns subﬁelds of

K and subgroups of K

containing

Nm(

K), the corollary is a restatement of the main theorem of Galois theory. The full

result is obtained by letting

K grow.

Remark 1.4. Corollary 1.3 also holds for archimedean local ﬁelds. The abelian

extensions of R are R and C , and their norm subgroups are R

and R

.LetH be a

subgroup of ﬁnite index in R

.ThenH ⊃ R

×m

for some m,andR

×m

= R

or R

according as m is odd or even (apply the intermediate value theorem). Therefore R

STATEMENTS OF THE MAIN THEOREMS 11

and R

are the only two subgroups of R

of ﬁnite index. Moreover, there is a unique

isomorphism (local Artin map in the real case)

→ Gal(C/R),

namely, that sending r to the identity map or complex conjugation according as r>0

or r<0. This veriﬁes the corollary for K = R, and it is even more obvious for K = C.

Remark 1.5. If K has characteristic zero, then every subgroup H of K

of ﬁnite

index is open. To prove this, observe that a subgroup H of ﬁnite index will contain

×m

for some m, and that Newton’s lemma (Math 676, 7.23) applied to X

− a

shows that any a ∈O

such that |1 − a| < |m|

is of the form u

with u ∈ 1+m.

Therefore H contains an open neighbourhood of 1 in K

, and, since it is a group,

this implies that it is open.

If K has characteristic p =0,thennot every subgroup of K

of ﬁnite index is

open. Note that an open subgroup is also closed (because its complement is a union

of cosets of the group, which are also open). Weil 1967, II.3, Proposition 10, shows

that 1 + m ≈



(product of copies of Z

indexed by N), from which it follows

that K

has a quotient isomorphic to



.Leta =(a

)

n∈

∈



,andlet

a(m)=(a(m)

)

n∈

be such that a(m)

= a

for n ≤ m and a(m)

=0otherwise.

Then a(m) → a as m →∞,andso⊕

is dense in



. Therefore any proper

subgroup of



containing ⊕

can not be closed. Because



/ ⊕

is a

vector space over F

, it will have subspaces of ﬁnite index, and the inverse image of

such a subspace in K

will be a nonclosed subgroup of ﬁnite index.

Remark 1.6. The composite of two ﬁnite unramiﬁed extensions of K is again

unramiﬁed, and therefore the union K

of all ﬁnite unramiﬁed extensions of K (in

) is an unramiﬁed extension of K. The residue ﬁeld

k of K

is an algebraic

closure of the residue ﬁeld k of K.

Every automorphism σ of K

ﬁxing K preserves the valuation |·|on K

,and

hence induces an automorphism ¯σ of

k/k.Themapσ → ¯σ is an isomorphism

Gal(K

/K) → Gal(

k/k). Therefore, there is a unique element Frob

∈ Gal(K

/K)

inducing the map x → x

k,andthemapα → Frob



Z → Gal(K

/K)isan

isomorphism of topological groups. Condition (a) of Theorem 1.1 can be re-stated as:

(a) for any prime element π of K, φ(π)actsasFrob

on K

Consequences of Theorems 1.1 and 1.2. Assume K possesses a local Artin

map, i.e., a homomorphism φ

: K

→ Gal(K

/K) with the properties (a) and (b)

of Theorem 1.1, and that Theorem 1.2 holds.

As we just remarked, for any prime element π of K, φ

(π)|K

=Frob

.If

u ∈ U

,thenπu is also a prime element, and so

(u)|K

= φ

(πu)|K

· φ

(π

−1

)|K

=id.

Therefore, U

is in the kernel of a → φ

(a)|K

: K

→ Gal(K

/K), and

(a)|K

=Frob

ord

(a)

for any a ∈ K

. In other words, the map φ

: K

→

Gal(K

/K) factors into

ord

−−→ Z

n→Frob

−−−−−→ Gal(K

/K).

12 I. LOCAL CLASS FIELD THEORY

If K

is the unramiﬁed extension of K of degree m,then

Nm(K

)={a ∈ K

| m|ord

(a)} = U

· π

because this is the kernel of a → φ

(a)|K

Choose a prime element π in K. Then any element a ∈ K

can be written uniquely

a = u · π

, u ∈ U

, m ∈ Z.Thus

≈ U

× Z,u· π

↔ (u, m).

Because U

is both open and closed in K

, this is an isomorphism of topological

groups (discrete topology on Z). The subgroups 1 + m

, n ≥ 0, form a fundamental

system of neighbourhoods of 1 in K. Clearly therefore, the subgroups (1 + m

) ×mZ

are open of ﬁnite index in U

×Z, and every open subgroup of ﬁnite index contains one

of this type. For a ﬁnite abelian extension L of K, the smallest f such that Nm(L

)

contains 1 + m

is called the conductor of L/K, except that, when Nm(L

) ⊃ U

the conductor is said to be 0. Thus L/K is unramiﬁed if and only if its conductor is

0, and it is tamely ramiﬁed if and only if its conductor is ≤ 1.

The homomorphisms

L/K

: K

/ Nm(L

) → Gal(L/K)

form a projective system as L runs through the ﬁnite abelian extensions of K, ordered

by inclusion. On passing to the limit, we obtain an isomorphism





) → Gal(K

/K)

—in particular, there is a one-to-one correspondence between the set of closed sub-

groups of



and the set of abelian extensions of K.Here



is the completion

of K

with respect to the topology for which the norm groups form a fundamental

system of neighbourhoods of 1. This topology on K

is called the norm topology.

According to Theorem 1.2, the norm groups are the open subgroups of ﬁnite index in

. The norm topology is coarser than the usual topology on K

—for example, U

is not open—but it induces the usual topology on U

. When we complete the terms

in the exact sequence

0 → U

→ K

ord

−−→ Z → 0

with respect to the norm topology, we obtain an exact sequence

0 → U

→



→



Z → 0.

Here



Z is completion of Z for the topology deﬁned by the subgroups of ﬁnite index.

Loosely speaking,



can be said to have been obtained from K

by completing Z.

(See Artin 1951, 9.3, for a detailed discussion of the norm topology on K

The choice of a prime element π determines a decomposition



= U

· π





into the product of two closed subgroups, and hence (by inﬁnite Galois theory),

a decomposition

= K

· K

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