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 fieldMathworldPlanetmath (that is, a finite extensionMathworldPlanetmath of the rational numbers ) and let 𝒪K be the ring of integersMathworldPlanetmath in K. The ring of integers of K is the analogue of in . As we know, enjoys the property that any number can be factored uniquely as a productPlanetmathPlanetmath of powers of primes. In particular, is a UFD and a PID (principal ideal domainMathworldPlanetmath). When is 𝒪K a UFD or a PID? This is a very hard question to answer. The ideal class groupPlanetmathPlanetmathPlanetmath and class number of K are objects that measures how far 𝒪K is from actually being a PID. In that sense, the class groups measure the arithmeticPlanetmathPlanetmath 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 groupMathworldPlanetmath of all fractional idealsMathworldPlanetmathPlanetmath 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 hK.

1.1 Basic Definitions

  1. 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 groupMathworldPlanetmath (the entry also discusses properties of ideal classes).

  2. 2.

    The Hilbert class fieldMathworldPlanetmath of K, usually denoted by H, is the maximal unramified abelian extensionMathworldPlanetmathPlanetmath of K. In particular, the Galois groupMathworldPlanetmath Gal(H/K) is isomorphicPlanetmathPlanetmathPlanetmath 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 characterizationsMathworldPlanetmath of H.

  3. 3.

    The concept of ray class group is a generalizationPlanetmathPlanetmath of the class group of a field. See also ray class field.

1.2 Computing Class Groups and Class Numbers

  1. 1.

    The class number formulaMathworldPlanetmath is one of the most important results in number theoryMathworldPlanetmathPlanetmath. It relates Dedekind zeta functions and class numbers (and other invariants of the field).

  2. 2.

    Minkowski’s theoremMathworldPlanetmath on lattices provides the well-known Minkowski’s constant, which in turn can be used to bound class numbers and discriminantsPlanetmathPlanetmathPlanetmath.

  3. 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 divisorsPlanetmathPlanetmath of hK and the unramified abelian extensions of K.

The following are theorems that explain the properties of class numbers in extensionsPlanetmathPlanetmathPlanetmathPlanetmath of number fields:

  1. 1.

    Class number divisibility in extensions: F/K Galois, [F:K] not divisible by p. Then p|hK implies p|hF.

  2. 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|hF then pf|hF for some f (see entry for details).

  3. 3.

    Extensions without unramified subextensions and class number divisibility: F/K such that there are no non-trivial abelian unramified subextensions. Then hK|hF.

  4. 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|hF then p|hK.

  5. 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|hF then p|hK.

1.4 Class Numbers of Cyclotomic Fields

Cyclotomic fieldsMathworldPlanetmath 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 ζn be a primitive nth root of unityMathworldPlanetmath. The field K=(ζn) is a cyclotomic field. We denote its class number by hn.

  1. 1.

    A prime numberMathworldPlanetmath p is said to be an irregular prime if hp is divisible by p (the entry on regular primes contains Kummer’s criterion for irregularity in terms of Bernoulli numbersMathworldPlanetmathPlanetmath). See some examples of regular primes.

  2. 2.

    Herbrand’s theorem relates Bernoulli numbers and certain subgroupsMathworldPlanetmathPlanetmath (or χ-components) of the ideal class group.

  3. 3.

    Stickelberger’s theorem on annihilatorsMathworldPlanetmathPlanetmath of the ideal class group of (ζp) (it also defines the Stickelberger elements).

  4. 4.

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

  5. 5.

    Vandiver’s conjecture states that a prime number p cannot divide the class number of the maximal real subfieldMathworldPlanetmath of (ζp).

  6. 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 (ζp).

1.5 Discriminants and Related Results

  1. 1.

    Definition of discriminant (http://planetmath.org/Discriminant) (also discusses the relationship with discriminants in other contexts).

  2. 2.

    A related concept: the root-discriminant.

  3. 3.

    Hermite’s theorem on extensions which are unramified outside a fixed set of primes.

References

  1. 1.

    Serge Lang, Algebraic Number TheoryMathworldPlanetmath. Springer-Verlag, New York.

  2. 2.

    Daniel A. Marcus, Number Fields, Springer, New York.

  3. 3.

    K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1998.

  4. 4.

    Lawrence C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.

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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