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topics on ideal class groups and discriminants (Topic)

Ideal Class Groups, Class Numbers and Discriminants (MSC 11R29)

Let $ K$ be a number field (that is, a finite extension of the rational numbers $ \mathbb{Q}$) and let $ \mathcal{O}_K$ be the ring of integers in $ K$. The ring of integers of $ K$ is the analogue of $ \mathbb{Z}$ in $ \mathbb{Q}$. As we know, $ \mathbb{Z}$ enjoys the property that any number can be factored uniquely as a product of powers of primes. In particular, $ \mathbb{Z}$ is a UFD and a PID (principal ideal domain). When is $ \mathcal{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 $ \mathcal{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, $ \operatorname{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 $ \vert\operatorname{Cl}(K)\vert$ is called the class number of $ K$ and it is usually written $ h_K$.

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 $ \operatorname{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 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.

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).

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\vert h_K$ implies $ p\vert 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\vert h_F$ then $ p^f\vert 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\vert h_F$.
  4. Class number divisibility in $ p$-extensions: $ F/K$ is a Galois $ p$-extension which is ramified at most at one prime. If $ p\vert h_F$ then $ p\vert 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\vert h_F$ then $ p\vert h_K$.

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 $ \zeta_n$ be a primitive $ n$th root of unity. The field $ K=\mathbb{Q}(\zeta_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 $ \chi$-components) of the ideal class group.
  3. Stickelberger's theorem on annihilators of the ideal class group of $ \mathbb{Q}(\zeta_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 $ \mathbb{Q}(\zeta_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 $ \mathbb{Q}(\zeta_p)$.

Discriminants and Related Results

  1. Definition of 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).



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See Also: ideal class, bibliography for number theory, class number divisibility in cyclic extensions, class number divisibility in $p$-extensions, class number formula, unramified extensions and class number divisibility, push-down theorem on class numbers, class number divisibility in extensions, using Minkowski's constant to find a class number, class number, existence of Hilbert class field, Minkowski's constant, examples of regular primes, irregular prime, Vandiver's conjecture, ray class group, algebraic number theory, an exact sequence for ray class groups, ideal classes form an abelian group

Also defines:  ideal class group
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Cross-references: algebraic number theory, fixed set, Hermite's theorem, root-discriminant, indexes of the group of cyclotomic units in the full unit group, maximal real subfield, Vandiver's conjecture, totally real fields, Thaine's theorem, Stickelberger elements, annihilators, Stickelberger's theorem, subgroups, Herbrand's theorem, examples of regular primes, Bernoulli numbers, terms, regular primes, irregular prime, prime number, root of unity, primitive, Fermat's last theorem, cyclotomic fields, totally ramified, push-down theorem on class numbers, abelian, extensions without unramified subextensions and class number divisibility, divide, cyclic, class number divisibility in cyclic extensions, implies, divisible, class number divisibility in extensions, extensions, prime divisors, connection, unramified extensions and class number divisibility, contains, using Minkowski's constant to find a class number, discriminants, bound, Minkowski's constant, Minkowski's theorem, invariants, Dedekind zeta functions, number theory, class number formula, ray class field, ray class group, characterizations, theory, field, class, link, isomorphic, Galois group, abelian extension, unramified, Hilbert class field, ideal classes form an abelian group, ideal class, size, fractional ideals, quotient group, class groups, measures, objects, class number, PID, UFD, primes, product, number, property, ring of integers, rational numbers, finite extension, number field
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This is version 10 of topics on ideal class groups and discriminants, born on 2005-03-10, modified 2008-01-22.
Object id is 6869, canonical name is ClassNumbersAndDiscriminantsTopicsOnClassGroups.
Accessed 5436 times total.

Classification:
AMS MSC: 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
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