代数数论基础与类域理论概览

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"这是一份关于代数数论的PDF文档，由J.S. Milne编写的版本3.08，日期为2020年7月19日。文档主要探讨了代数数论的基本概念，包括代数数域、代数数、理想、单位元以及唯一分解性质等，并预备介绍类域理论。此外，还提到了文档的更新历史，如v2.01至v3.01的修改内容和页数。" 正文: 代数数论是数学的一个重要分支，它研究的是与有理数Q的有限扩张相关的算术性质。这里的"有限扩张"指的是代数数域，即包含有理数并能通过有限次根式运算得到的所有数的集合。一个代数数就是代数数域中的元素，这些数满足某个整系数多项式的方程。在代数数论中，我们关注的核心对象是代数数域内的整数环，也就是该域中所有可以表示为多项式根的整数。例如，对于实数域的扩展$\mathbb{Q}(\sqrt{2})$，整数环将包括所有形式为$a+b\sqrt{2}$的数，其中a和b是整数。这个环的算术特性，如是否存在唯一分解定理（即每个非零非单位元素是否能唯一地分解为素因子的乘积），是代数数论研究的关键问题。另一个重要的概念是理想，它是环中的一个子集，满足特定的封闭性质。在整数环中，理想可以被看作是某些元素生成的集合，比如所有形如ma+nb的数，其中m和n是任意整数，a和b是理想的生成元。理想的性质，比如它们的交集、商集以及与环中元素的关系，对理解代数数域的结构至关重要。类域理论是代数数论的一个高级分支，它涉及代数数域的阿贝尔扩张，即其伽罗瓦群是交换群的扩张。伽罗瓦理论提供了理解和分类代数扩张的方法，而类域理论则进一步将这些阿贝尔扩张与数域自身的算术性质联系起来。它旨在用数域的内在属性来描述所有阿贝尔扩张，从而提供了一种深刻的统一视角。 J.S. Milne的这份文档显然涵盖了这些基本概念，并可能深入到更复杂的主题，如高斯整数、克罗内克-韦伯定理、类群和理想类群、黎曼猜想等。通过解决练习和参考索引，学习者可以逐步构建对代数数论的深入理解。随着文档版本的更新，作者修复了错误，增加了练习和索引，使得内容更加完善，学习者可以从中受益匪浅。特别是对于那些对数论感兴趣，特别是对代数数论及其与类域理论关系有探索欲望的读者，这份文档无疑是一个宝贵的资源。

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CHAPTER 1

Preliminaries from Commutative

Algebra

Many results that were ﬁrst proved for rings of integers in number ﬁelds are true for more

general commutative rings, and it is more natural to prove them in that context.

Basic deﬁnitions

All rings will be commutative, and have an identity element (i.e., an element

such that

1a D a

for all

a 2 A

), and a homomorphism of rings will map the identity element to the

identity element.

A ring

together with a homomorphism of rings

A ! B

will be referred to as an

algebra

. We use this terminology mainly when

is a subring of

. In this case, for

elements

;:::; ˇ

, we let

AŒˇ

;:::; ˇ



denote the smallest subring of

containing

and the

. It consists of all polynomials in the

with coefﬁcients in

, i.e., elements of

the form

:::i

:::ˇ

; a

:::i

2 A:

We also refer to

AŒˇ

;:::; ˇ



as the

-subalgebra of

B generated

by the

, and when

B D AŒˇ

;:::; ˇ

 we say that the ˇ

generate B as an A-algebra.

For elements

;: : :

, we let

;: : :/

denote the smallest ideal containing the

. It consists of ﬁnite sums

, and it is called the

ideal generated by a

;: : :

When a and b are ideals in A, we deﬁne

a Cb D fa Cb j a 2 a, b 2 bg:

It is again an ideal in

— in fact, it is the smallest ideal containing both

and

. If

a D .a

;:::; a

/ and b D .b

;:::; b

/, then a Cb D .a

;:::; a

;:::; b

Given an ideal

, we can form the quotient ring

A=a

. Let

f WA ! A=a

be the

homomorphism

a 7! a Ca

; then

b 7! f

1

.b/

deﬁnes a one-to-one correspondence between

the ideals of A=a and the ideals of A containing a, and

A=f

1

.b/ ' .A=a/=b:

See also the notes A Primer of Commutative Algebra available on my website.

Ideals in products of rings 15

A proper ideal

prime

ab 2 a ) a

b 2 a

. An ideal

is prime if and only if

the quotient ring

A=a

is an integral domain. A nonzero element



is said to be

prime

./ is a prime ideal; equivalently, if jab ) ja or jb.

An ideal

maximal

if it is maximal among the proper ideals of

, i.e., if

m ¤ A

and there does not exist an ideal

a ¤ A

containing

but distinct from it. An ideal

maximal if and only if

A=a

is a ﬁeld. Every proper ideal

is contained in a maximal

ideal — if

is noetherian (see below) this is obvious; otherwise the proof requires Zorn’s

lemma. In particular, every nonunit in A is contained in a maximal ideal.

There are the implications:

is a Euclidean domain

) A

is a principal ideal domain

) A is a unique factorization domain (see any good graduate algebra course).

Ideals in products of rings

PROPOSITION 1.1

Consider a product of rings

A B

. If

and

are ideals in

and

respectively, then

a b

is an ideal in

A B

, and every ideal in

A B

is of this form. The

prime ideals of A B are the ideals of the form

p B (p a prime ideal of A), A p (p a prime ideal of B).

PROOF. Let c be an ideal in A B, and let

a D fa 2 A j .a;0/ 2 cg; b D fb 2 B j .0; b/ 2 cg:

Clearly

a b  c

. Conversely, let

.a; b/ 2 c

. Then

.a; 0/ D .a;b/ .1; 0/ 2 c

and

.0;b/ D

.a; b/ .0;1/ 2 c, and so .a;b/ 2 a b:

Recall that an ideal c  C is prime if and only if C=c is an integral domain. The map

A B ! A=a B=b; .a;b/ 7! .a Ca;b Cb/

has kernel a b, and hence induces an isomorphism

.A B/=.a b/ ' A=a B=b:

Now use that a product of rings is an integral domain if and only if one ring is zero and the

other is an integral domain.

REMARK 1.2

The lemma extends in an obvious way to a ﬁnite product of rings: the ideals

A

are of the form

a

with

an ideal in

; moreover,

a

is prime if and only if there is a

such that

is a prime ideal in

and

D A

for

i ¤ j:

Noetherian rings

A ring A is noetherian if every ideal in A is ﬁnitely generated.

PROPOSITION 1.3 The following conditions on a ring A are equivalent:

(a) A is noetherian.

(b) Every ascending chain of ideals

 a

   a

 

eventually becomes constant, i.e., for some n, a

D a

nC1

D .

16 1. Preliminaries from Commutative Algebra

(c)

Every nonempty set

of ideals in

has a maximal element, i.e., there exists an ideal

in S not properly contained in any other ideal in S.

PROOF.

(a)

)

(b): Let

a D

; it is an ideal, and hence is ﬁnitely generated, say

a D

;: : : ; a

/. For some n, a

will contain all the a

, and so a

D a

nC1

D  D a.

(b))(c): Let a

2 S. If a

is not a maximal element of S, then there exists an a

2 S such

that

& a

. If

is not maximal, then there exists an

etc.. From (b) we know that this

process will lead to a maximal element after only ﬁnitely many steps.

(c)

)

(a): Let

be an ideal in

, and let

be the set of ﬁnitely generated ideals contained in

Then

is nonempty because it contains the zero ideal, and so it contains a maximal element,

say,

D .a

;: : : ; a

. If

¤ a

, then there exists an element

a 2 a ra

, and

;: : : ; a

;a/

will be a ﬁnitely generated ideal in

properly containing

. This contradicts the deﬁnition

of a

A famous theorem of Hilbert states that

kŒX

;:::; X



is noetherian. In practice, almost all

the rings that arise naturally in algebraic number theory or algebraic geometry are noetherian,

but not all rings are noetherian. For example, the ring

kŒX

;: : : ; X

;: : :

of polynomials in

an inﬁnite sequence of symbols is not noetherian because the chain of ideals

/  .X

/  

never becomes constant.

PROPOSITION 1.4

Every nonzero nonunit element of a noetherian integral domain can be

written as a product of irreducible elements.

PROOF. We shall need to use that, for elements a and b of an integral domain A,

.a/  .b/ ” bja, with equality if and only if b D a unit:

The ﬁrst assertion is obvious. For the second, note that if

a D bc

and

b D ad

then

a D bc D

adc, and so dc D 1. Hence both c and d are units.

Suppose that the statement of the proposition is false for a noetherian integral domain

Then there exists an element

a 2 A

which contradicts the statement and is such that

.a/

maximal among the ideals generated by such elements (here we use that

is noetherian).

Since

cannot be written as a product of irreducible elements, it is not itself irreducible,

and so

a D bc

with

and

nonunits. Clearly

.b/  .a/

, and the ideals can’t be equal for

otherwise

would be a unit. From the maximality of

.a/

, we deduce that

can be written

as a product of irreducible elements, and similarly for

. Thus

is a product of irreducible

elements, and we have a contradiction.

REMARK 1.5

Note that the proposition fails for the ring

of all algebraic integers in the

algebraic closure of

, because, for example, we can keep extracting square roots —

an algebraic integer

cannot be an irreducible element of

because

will also be an

algebraic integer and ˛ D

˛ 

˛. Thus O is not noetherian.

Noetherian modules

Let

be a ring. An

-module

is said to be

noetherian

if every submodule is ﬁnitely

generated.

Local rings 17

PROPOSITION 1.6 The following conditions on an A-module M are equivalent:

(a) M is noetherian;

(b) every ascending chain of submodules eventually becomes constant;

PROOF. Similar to the proof of Proposition 1.3.

PROPOSITION 1.7

Let

be an

-module, and let

be a submodule of

. If

and

M=N are both noetherian, then so also is M .

PROOF.

I claim that if

 M

are submodules of

such that

\N D M

and

have the same image in

M=N

, then

D M

. To see this, let

x 2 M

; the

second condition implies that there exists a

y 2 M

with the same image as

M=N

, i.e.,

such that x y 2 N . Then x y 2 M

\N  M

, and so x 2 M

Now consider an ascending chain of submodules of

. If

M=N

is noetherian, the image

of the chain in

M=N

becomes constant, and if

is noetherian, the intersection of the chain

with N becomes constant. Now the claim shows that the chain itself becomes constant.

PROPOSITION 1.8

Let

be a noetherian ring. Then every ﬁnitely generated

-module is

noetherian.

PROOF.

is generated by a single element, then

M  A=a

for some ideal

, and

the statement is obvious. We argue by induction on the minimum number

of generators of

. Since

contains a submodule

generated by

n 1

elements such that the quotient

M=N is generated by a single element, the statement follows from Proposition 1.7.

Local rings

A ring

is said to be

local

if it has exactly one maximal ideal

. In this case,



D A r m

(complement of m in A).

LEMMA 1.9 (NAKAYAMA’S LEMMA)

Let

be a local ring and

a proper ideal in

. Let

M be a ﬁnitely generated A-module, and deﬁne

aM D

j a

2 a; m

2 M

(a) If aM D M , then M D 0:

(b) If N is a submodule of M such that N CaM D M , then N D M:

PROOF.

(a) Suppose that

aM D M

but

M ¤ 0

. Choose a minimal set of generators

;: : : ; e

g for M , n  1. As e

2 M D aM , there exist a

2 a such that

D a

CCa

Then

.1 a

D a

CCa

1 a

is not in

, it is a unit, and so

;:::; e

generates

, which contradicts our

choice of fe

;: : : ; e

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