algebraic number theory

algebraic number theory

Algebraic Number Theory

This entry is a of to entries on

algebraic number theory

in Planetmath (therefore to be always

under construction

). It is the hope of the author(s) that someday this can be used as a “graduate text” to learn the subject by reading the

individual

entries listed here. Each a brief description of the

concepts

, which is

expanded

in the entries. Some of the concepts might be missing in Planetmath as of today (please consider writing an entry on them!). In to organize the entry in

sections

, we followed the main reference

[

Mar

]

.

1

Introduction

The entry

number theory

contains a nice

introduction

to the broad subject. From very early on, mathematicians have tried to understand the

integer

solutions

of

polynomial equations

(e.g.

Pythagorean triples

). One of the main motivational examples for the subject is Fermat’s Last

Theorem

(when does

x

n

+

y

n

=

z

n

have integer solutions?). The study of integer solutions immediately leads to the study of

algebraic numbers

(see

y

2

=

x

3

-

2

(

http://planetmath.org/Y2X32

) for an example). Algebraic number theory is the study of algebraic numbers, their properties and their applications.

•

As an introduction, the reader should be comfortable with the basic

theory of rational and irrational numbers

, and its complementary entry, the basic

theory of algebraic and transcendental numbers

.

2

Number Fields and Rings of Integers

1.

The main object of study in algebraic number theory is the

number field

. A number field

K

is a

finite field extension

of

ℚ

. Since a

finite extension

of fields is an

algebraic extension

,

K

/

ℚ

is

algebraic

. Thus, every

α

∈

K

is an algebraic number (

http://planetmath.org/AlgebraicNumber

).

2.

The

ring of integers

of

K

, usually denoted by

𝒪

K

, is the set of all

algebraic integers

of

K

.

𝒪

K

is a

commutative ring

with

identity

(

http://planetmath.org/Unity

). See

examples of ring

of integers of a number field.

3.

Real and complex embeddings

of a number field. Read also about totally real and

imaginary

fields.

4.

Norm and trace of an algebraic number. See also this entry (

http://planetmath.org/NormAndTraceOfAlgebraicNumber

). One also can take the norm of an ideal.

5.

The

discriminant

of a number field measures the

ramification

of the field (read the following section for more details on ramification).

6.

The ring of integers of a number field is

finitely generated

over

ℤ

(

http://planetmath.org/RingOfIntegersOfANumberFieldIsFinitelyGeneratedOverMathbbZ

).

7.

Euclidean number fields.

3

Decomposition of Prime Ideals

1.

It is a well-known fact that the ring of integers of a number field is a

Dedekind domain

.

2.

Every non-zero

fractional ideal

in a Dedekind domain is

invertible

. In fact, the set of all non-zero fractional ideals forms a group under

multiplication

(see also Prüfer ring and

multiplication ring

).

3.

Notice that the ring of integers

𝒪

K

of a number field is not necessarily a PID nor a UFD (see example of ring which is not a UFD). However, every fractional ideal in a Dedekind domain factors uniquely as a

product

of powers of

prime ideals

. In particular, the ideals of

𝒪

K

factor uniquely as a product of prime ideals.

4.

Let

F

/

K

be an

extension

of number fields. Let

𝔭

be a prime ideal of

𝒪

K

, then

𝔭

⁢

𝒪

F

is an ideal of

F

. What is the

factorization

of

𝔭

⁢

𝒪

F

into prime ideals of

F

? Read about

splitting and ramification in number fields and Galois extensions

for a detailed explanation and definitions of the terminology.

5.

In to understand ramification in a more general setting, read ramify (

http://planetmath.org/Ramify

),

inertia group

and

decomposition group

.

6.

See the entry

ramification of archimedean places

for the case of infinite places.

7.

An important example: prime ideal decomposition in

quadratic extensions

of

ℚ

(

http://planetmath.org/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ

).

8.

Another important case: prime ideal decomposition in

cyclotomic extensions

of

ℚ

(

http://planetmath.org/PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ

).

9.

Explicit

examples of prime ideal decomposition in number fields

.

10.

More generally, read calculating the splitting of primes.

4

Ideal Class Groups

The

ideal class group

Cl

⁡

(

K

)

of a number field

K

is the

quotient group

of all fractional ideals modulo principal fractional ideals. In some sense, it measures the

arithmetic

complexity of the number field (how far

K

is from being a PID). The

class number

of

K

, denoted by

h

K

, is the of

Cl

⁡

(

K

)

. See

topics on ideal class groups and discriminants

for a detailed exposition.

5

The Unit Group

The

unit group

of a number field

K

is the group of units of the ring of integers

𝒪

K

, and it is usually denoted by

𝒪

K

×

.

1.

The

structure

of the unit group is described by Dirichlet’s unit theorem, which asserts the

existence

of a system of

fundamental units

.

2.

An application of Dirichlet’s unit theorem:

units of quadratic fields

.

3.

The

regulator

is an important

invariant

of the unit group (it appears in the

class number formula

(

http://planetmath.org/ClassNumberFormula

)).

4.

The

cyclotomic units

are a

subgroup

of the group of units of a

cyclotomic field

with very interesting properties. The

cyclotomic units are algebraic units

.

6

Zeta Functions and

L

-functions

1.

The prototype of

zeta function

is

ζ

⁢

(

s

)

, the

Riemann zeta function

(

http://planetmath.org/RiemannZetaFunction

) (the entry also discusses the famous

Riemann hypothesis

).

2.

More generally, for every number field

K

one can define a

Dedekind zeta function

ζ

K

⁢

(

s

)

.

3.

The Dedekind zeta function of a number field

satisfies

the so-called class number formula, which relates many of the invariants of the number field.

7

Class Field Theory

Class field theory

studies the

abelian extensions

of number fields.

1.

The

Kronecker-Weber theorem

classifies the possible abelian extensions of

ℚ

.

2.

The abelian extensions of quadratic imaginary number fields can be described using

elliptic curves

with

complex multiplication

.

3.

The

Artin map

is an important tool in class field theory. Class field theory and the Artin map can be presented in of idèles (

http://planetmath.org/Idele

) and adèles (

http://planetmath.org/Adele

).

4.

The

Hilbert class field

H

of a number field

K

is the

maximal

unramified abelian extension of

K

. The key property of

H

is that the

Galois group

Gal

⁡

(

H

/

K

)

is

isomorphic

to the ideal class group

Cl

⁡

(

K

)

.

5.

Ray class fields

are maximal abelian extensions with

conductor

. See also

ray class groups

.

8

Local Fields

Many problems in number theory can be treated “locally” or one prime at a . For this, one works over

local fields

, like

ℚ

p

or the

completion

of a number field at a prime

𝔓

.

1.

Definition of local field (

http://planetmath.org/LocalField

).

2.

The main example and motivation: the

p

-adic

rationals

and the

p

-adic integers (

http://planetmath.org/PAdicIntegers

) (see also

p

-adic

valuation

(

http://planetmath.org/PAdicValuation

)).

3.

Let

v

be a valuation of the field

K

(see the entry valuation (

http://planetmath.org/Valuation

) for a comprehensive introduction). The completion (

http://planetmath.org/Completion

) of

K

with respect to

v

is a local field. For example,

ℚ

p

is the completion of

ℚ

with respect to the

p

-adic valuation (

http://planetmath.org/PAdicValuation

).

4.

Read also about

discrete valuation rings

.

5.

Hensel’s lemma provides a criterion to prove the existence of roots of

polynomials

in local fields. See also examples for Hensel’s lemma.

9

Galois Representations

1.

Recall that a number field is a finite extension of

ℚ

. We can also study

infinite

extensions. Read about

infinite Galois theory

.

2.

Some number theorists would say that algebraic number theory is the study of the

absolute Galois group

of

ℚ

,

Gal

⁡

(

ℚ

¯

/

ℚ

)

.

3.

In order to understand

Gal

⁡

(

ℚ

¯

/

ℚ

)

, one studies

Galois representations

(the entry is an excellent overview and introduction to Galois representation theory).

10

Elliptic Curves

Elliptic curves are, essentially, equations of the form

y

2

=

x

3

+

A

⁢

x

+

B

. Read the entry on the

arithmetic of elliptic curves

for a full account of this beautiful theory.

11

Modular Forms

1.

Definition of

modular form

and the

Hecke algebra

of

Hecke operators

.

References

Mar

Daniel A. Marcus,

Number Fields

, Springer, New York.

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

Title

algebraic number theory

Canonical name

AlgebraicNumberTheory

Date of creation

2013-03-22 15:08:05

Last modified on

2013-03-22 15:08:05

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

34

Author

alozano (2414)

Entry type

Topic

Classification

msc 11S99

Classification

msc 11R99

Classification

msc 11-01

Related topic

NormAndTraceOfAlgebraicNumber

Related topic

ModularForms

Related topic

HeckeOperator

Related topic

BibliographyForNumberTheory

Related topic

ArithmeticOfEllipticCurves

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Related topic

ExamplesOfRingOfIntegersOfANumberField

Related topic

NumberField

Related topic

TheoryOfAlgebraicNumbers

Related topic

TheoryOfRa

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