topics on ideal class groups and discriminants
Ideal Class Groups, Class Numbers and Discriminants (http://planetmath.org/browse/objects/11R29/MSC 11R29)
Let be a number field (that is, a finite extension of the rational numbers ) and let be the ring of integers in . The ring of integers of is the analogue of in . As we know, enjoys the property that any number can be factored uniquely as a product of powers of primes. In particular, is a UFD and a PID (principal ideal domain). When is a UFD or a PID? This is a very hard question to answer. The ideal class group and class number of are objects that measures how far 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:
1.1 Basic Definitions
The Hilbert class field of , usually denoted by , is the maximal unramified abelian extension of . In particular, the Galois group is isomorphic to the class group of 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 .
1.2 Computing Class Groups and Class Numbers
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 and the unramified abelian extensions of .
The following are theorems that explain the properties of class numbers in extensions of number fields:
Class number divisibility in cyclic extensions: Galois and cyclic with not divisible by and does not divide the class number of intermediate extensions. Then if then for some (see entry for details).
Extensions without unramified subextensions and class number divisibility: such that there are no non-trivial abelian unramified subextensions. Then .
Class number divisibility in -extensions (http://planetmath.org/ClassNumberDivisibilityInPExtensions): is a Galois -extension which is ramified at most at one prime. If then .
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 , let be a primitive th root of unity. The field is a cyclotomic field. We denote its class number by .
Thaine’s theorem is the counterpart of Stickelberger’s theorem for totally real fields.
1.5 Discriminants and Related Results
Serge Lang, Algebraic Number Theory. Springer-Verlag, New York.
Daniel A. Marcus, Number Fields, Springer, New York.
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1998.
Lawrence C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.
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|Title||topics on ideal class groups and discriminants|
|Date of creation||2013-03-22 15:07:44|
|Last modified on||2013-03-22 15:07:44|
|Last modified by||alozano (2414)|
|Defines||ideal class group|