algebraic number theory


Algebraic Number Theory

This entry is a of to entries on algebraic number theoryMathworldPlanetmath 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 sectionsPlanetmathPlanetmathPlanetmath, we followed the main reference [Mar].

1 Introduction

The entry number theoryMathworldPlanetmath 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 triplesMathworldPlanetmath). One of the main motivational examples for the subject is Fermat’s Last Theorem (when does xn+yn=zn have integer solutions?). The study of integer solutions immediately leads to the study of algebraic numbersMathworldPlanetmath (see y2=x3-2 (http://planetmath.org/Y2X32) for an example). Algebraic number theory is the study of algebraic numbers, their properties and their applications.

2 Number Fields and Rings of Integers

  1. 1.

    The main object of study in algebraic number theory is the number fieldMathworldPlanetmath. A number field K is a finite field extension of . Since a finite extension of fields is an algebraic extensionMathworldPlanetmath, K/ is algebraicMathworldPlanetmathPlanetmath. Thus, every αK is an algebraic number (http://planetmath.org/AlgebraicNumber).

  2. 2.

    The ring of integersMathworldPlanetmath of K, usually denoted by 𝒪K, is the set of all algebraic integersMathworldPlanetmath of K. 𝒪K is a commutative ring with identityPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Unity). See examples of ring of integers of a number field.

  3. 3.

    Real and complex embeddings of a number field. Read also about totally real and imaginaryPlanetmathPlanetmath fields.

  4. 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. 5.

    The discriminantPlanetmathPlanetmathPlanetmathPlanetmath of a number field measures the ramification of the field (read the following section for more details on ramification).

  6. 6.

    The ring of integers of a number field is finitely generatedMathworldPlanetmathPlanetmathPlanetmath over (http://planetmath.org/RingOfIntegersOfANumberFieldIsFinitelyGeneratedOverMathbbZ).

  7. 7.

    Euclidean number fields.

3 Decomposition of Prime Ideals

  1. 1.

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

  2. 2.

    Every non-zero fractional idealMathworldPlanetmathPlanetmath in a Dedekind domain is invertible. In fact, the set of all non-zero fractional ideals forms a group under multiplicationPlanetmathPlanetmath (see also Prüfer ring and multiplication ring).

  3. 3.

    Notice that the ring of integers 𝒪K 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 productMathworldPlanetmath of powers of prime idealsMathworldPlanetmathPlanetmath. In particular, the ideals of 𝒪K factor uniquely as a product of prime ideals.

  4. 4.

    Let F/K be an extensionPlanetmathPlanetmathPlanetmathPlanetmath of number fields. Let 𝔭 be a prime ideal of 𝒪K, then 𝔭𝒪F is an ideal of F. What is the factorization of 𝔭𝒪F 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. 5.

    In to understand ramification in a more general setting, read ramify (http://planetmath.org/Ramify), inertia group and decomposition group.

  6. 6.

    See the entry ramification of archimedean places for the case of infinite places.

  7. 7.

    An important example: prime ideal decomposition in quadratic extensions of (http://planetmath.org/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ).

  8. 8.

    Another important case: prime ideal decomposition in cyclotomic extensions of (http://planetmath.org/PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ).

  9. 9.
  10. 10.

    More generally, read calculating the splitting of primes.

4 Ideal Class Groups

The ideal class groupPlanetmathPlanetmathPlanetmath Cl(K) of a number field K is the quotient groupMathworldPlanetmath 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 hK, 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 𝒪K, and it is usually denoted by 𝒪K×.

  1. 1.

    The structureMathworldPlanetmath of the unit group is described by Dirichlet’s unit theorem, which asserts the existence of a system of fundamental unitsMathworldPlanetmath.

  2. 2.

    An application of Dirichlet’s unit theorem: units of quadratic fields.

  3. 3.

    The regulatorMathworldPlanetmath is an important invariant of the unit group (it appears in the class number formulaMathworldPlanetmath (http://planetmath.org/ClassNumberFormula)).

  4. 4.

    The cyclotomic units are a subgroupMathworldPlanetmathPlanetmath of the group of units of a cyclotomic fieldMathworldPlanetmath with very interesting properties. The cyclotomic units are algebraic units.

6 Zeta Functions and L-functions

  1. 1.

    The prototype of zeta function is ζ(s), the Riemann zeta functionMathworldPlanetmath (http://planetmath.org/RiemannZetaFunction) (the entry also discusses the famous Riemann hypothesis).

  2. 2.

    More generally, for every number field K one can define a Dedekind zeta function ζK(s).

  3. 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 extensionsMathworldPlanetmathPlanetmath of number fields.

  1. 1.

    The Kronecker-Weber theoremMathworldPlanetmath classifies the possible abelian extensions of .

  2. 2.

    The abelian extensions of quadratic imaginary number fields can be described using elliptic curvesMathworldPlanetmath with complex multiplicationMathworldPlanetmath.

  3. 3.

    The Artin mapMathworldPlanetmath 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. 4.

    The Hilbert class fieldMathworldPlanetmath H of a number field K is the maximal unramified abelian extension of K. The key property of H is that the Galois groupMathworldPlanetmath Gal(H/K) is isomorphicPlanetmathPlanetmathPlanetmath to the ideal class group Cl(K).

  5. 5.

    Ray class fields are maximal abelian extensions with conductorPlanetmathPlanetmath. 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 fieldsMathworldPlanetmath, like p or the completion of a number field at a prime 𝔓.

  1. 1.

    Definition of local field (http://planetmath.org/LocalField).

  2. 2.

    The main example and motivation: the p-adic rationals and the p-adic integers (http://planetmath.org/PAdicIntegers) (see also p-adic valuationMathworldPlanetmathPlanetmath (http://planetmath.org/PAdicValuation)).

  3. 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, p is the completion of with respect to the p-adic valuation (http://planetmath.org/PAdicValuation).

  4. 4.

    Read also about discrete valuation rings.

  5. 5.

    Hensel’s lemma provides a criterion to prove the existence of roots of polynomialsMathworldPlanetmathPlanetmath in local fields. See also examples for Hensel’s lemma.

9 Galois Representations

  1. 1.

    Recall that a number field is a finite extension of . We can also study infiniteMathworldPlanetmathPlanetmath extensions. Read about infinite Galois theory.

  2. 2.

    Some number theorists would say that algebraic number theory is the study of the absolute Galois group of , Gal(¯/).

  3. 3.

    In order to understand Gal(¯/), one studies Galois representationsMathworldPlanetmath (the entry is an excellent overview and introduction to Galois representation theory).

10 Elliptic Curves

Elliptic curves are, essentially, equations of the form y2=x3+Ax+B. Read the entry on the arithmetic of elliptic curves for a full account of this beautiful theory.

11 Modular Forms

  1. 1.

    Definition of modular formMathworldPlanetmath and the Hecke algebraMathworldPlanetmath 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