topics on ideal class groups and discriminants

topics on ideal class groups and discriminants

Ideal Class Groups, Class Numbers and Discriminants (

http://planetmath.org/browse/objects/11R29/

MSC 11R29)

Let

K

be a

number field

(that is, a

finite extension

of the

rational numbers

ℚ

) and let

𝒪

K

be the

ring of integers

in

K

. The ring of integers of

K

is the analogue of

ℤ

in

ℚ

. As we know,

ℤ

enjoys the

property

that any number can be factored uniquely as a

product

of powers of primes. In particular,

ℤ

is a UFD and a PID (

principal ideal domain

). When is

𝒪

K

a UFD or a PID? This is a very hard question to answer. The

ideal class group

and

class number

of

K

are objects that measures how far

𝒪

K

is from actually being a PID. In that sense, the

class groups

measure the

arithmetic

complexity of a number field. We include the basic

definition

of class group here for convenience of the reader:

Definition 1

.

The class group,

Cl

⁡

(

K

)

, of a number field

K

is defined to be the

quotient group

of all

fractional ideals

of

K

modulo principal fractional ideals. The size of the class group

|

Cl

⁡

(

K

)

|

is called the class number of

K

and it is usually written

h

K

.

1.1

Basic Definitions

1.

The definition of class group and class number can be found at the entry

ideal class

. Notice that the ideal classes form an

abelian group

(the entry also discusses properties of ideal classes).

2.

The

Hilbert class field

of

K

, usually denoted by

H

, is the

maximal

unramified

abelian extension

of

K

. In particular, the

Galois group

Gal

⁡

(

H

/

K

)

is

isomorphic

to the class group of

K

which is the link between ramification,

class field theory

and class numbers. The entry on the

existence

of the Hilbert class field (

http://planetmath.org/ExistenceOfHilbertClassField

) discusses alternative

characterizations

of

H

.

3.

The

concept

of

ray class group

is a

generalization

of the class group of a field. See also

ray class field

.

1.2

Computing Class Groups and Class Numbers

1.

The

class number formula

is one of the most important results in

number theory

. It relates

Dedekind zeta functions

and class numbers (and other

invariants

of the field).

2.

Minkowski’s

theorem

on lattices provides the well-known Minkowski’s

constant

, which in turn can be used to bound class numbers and

discriminants

.

3.

Using Minkowski’s constant to find a class number (contains examples).

1.3

Divisibility Properties of Class Numbers

The entry on

unramified extensions and class number divisibility

is a corollary of the existence of the Hilbert class field and clarifies the connection between the

prime divisors

of

h

K

and the unramified abelian extensions of

K

.

The following are theorems that explain the properties of class numbers in

extensions

of number fields:

1.

Class number divisibility in extensions

:

F

/

K

Galois,

[

F

:

K

]

not

divisible

by

p

. Then

p

|

h

K

implies

p

|

h

F

.

2.

Class number divisibility in cyclic extensions

:

F

/

K

Galois and cyclic with

[

F

:

K

]

not divisible by

p

and

p

does not divide the class number of intermediate extensions. Then if

p

|

h

F

then

p

f

|

h

F

for some

f

(see entry for details).

3.

Extensions without unramified subextensions and class number divisibility

:

F

/

K

such that there are no non-trivial

abelian

unramified subextensions. Then

h

K

|

h

F

.

4.

Class number

divisibility

in

p

-extensions (

http://planetmath.org/ClassNumberDivisibilityInPExtensions

):

F

/

K

is a Galois

p

-extension which is ramified at most at one prime. If

p

|

h

F

then

p

|

h

K

.

5.

Push-down theorem on class numbers

:

F

/

K

is a

p

-extension which is ramified exactly at one prime and this prime is

totally ramified

. If

p

|

h

F

then

p

|

h

K

.

1.4

Class Numbers of Cyclotomic Fields

Cyclotomic fields

have been the object of extensive study. For example, they are crucial in some of the “easy” cases of Fermat’s Last Theorem. For any number

n

, let

ζ

n

be a

primitive

n

th

root of unity

. The field

K

=

ℚ

⁢

(

ζ

n

)

is a cyclotomic field. We denote its class number by

h

n

.

1.

A

prime number

p

is said to be an

irregular prime

if

h

p

is divisible by

p

(the entry on

regular primes

contains Kummer’s criterion for irregularity in terms of

Bernoulli numbers

). See some

examples of regular primes

.

2.

Herbrand’s theorem relates Bernoulli numbers and certain

subgroups

(or

χ

-

components

) of the ideal class group.

3.

Stickelberger’s theorem on

annihilators

of the ideal class group of

ℚ

⁢

(

ζ

p

)

(it also defines the

Stickelberger elements

).

4.

Thaine’s theorem is the counterpart of Stickelberger’s theorem for totally real fields.

5.

Vandiver’s

conjecture

states that a prime number

p

cannot divide the class number of the maximal real

subfield

of

ℚ

⁢

(

ζ

p

)

.

6.

The index of the group of

cyclotomic units

in the full

unit groups

is exactly the class number of the maximal real subfield of

ℚ

⁢

(

ζ

p

)

.

1.5

Discriminants and Related Results

1.

Definition of discriminant (

http://planetmath.org/Discriminant

) (also discusses the relationship with discriminants in other contexts).

2.

A related concept: the

root-discriminant

.

3.

Hermite’s theorem on extensions which are unramified outside a

fixed set

of primes.

References

1.

Serge Lang,

Algebraic Number Theory

. Springer-Verlag, New York.

2.

Daniel A. Marcus,

Number Fields

, Springer, New York.

3.

K. Ireland, M. Rosen,

A Classical

Introduction

to Modern Number Theory

, Springer-Verlag, 1998.

4.

Lawrence C. Washington,

Introduction to Cyclotomic Fields

, Springer-Verlag, New York.

Note: If you would like to contribute to this entry, please send an email to the author (alozano).

Title

topics on ideal class groups and discriminants

Canonical

name

TopicsOnIdealClassGroupsAndDiscriminants

Date of creation

2013-03-22 15:07:44

Last modified on

2013-03-22 15:07:44

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

13

Author

alozano (2414)

Entry type

Topic

Classification

msc 11R29

Related topic

IdealClass

Related topic

BibliographyForNumberTheory

Related topic

ClassNumberDivisibilityInCyclicExtensions

Related topic

ClassNumberDivisibilityInPExtensions

Related topic

ClassNumberFormula

Related topic

UnramifiedExtensionsAndClassNumberDivisibility

Related topic

PushDownTheoremOnClassNumbers

Related topic

ClassNumberDivisibilityInExtension

Defines

ideal class group

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