Principal ideal theorem class field theory pdf

In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for. Then any element in the ray class group modulo m of k becomes trivial in the ray class group modulo n of l. With his introduction of ideles he was able to give a natural formulation of class. Density of the prime ideals splitting in an extension. On the principal ideal theorem in arithmetic topology. Every ideal of an imaginary quadratic field k is represented by a number in the field. In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of the term also has another, similar meaning in order theory, where it refers to an order ideal in a poset generated by a single element which is. These notes are based on a course in class field theory given by. Intro to class field theory and the chebotarev theorem lucas lingle august 28, 2014 abstract this paper is an exposition on class eld theory. Splitting modules and the principal ideal theorem 7 chapter xiv. For every abelian extension of number fields lk there exists an okideal f such that all primes of k that are principal. Notes on class field theory updated 17 mar 2017 kiran s.

Soon after takagis fundamental papers, there arose the question whether algebraic function elds with nite base eld could be treated similarly, i. A type of principal ideal theorem of class field theory. Class field theory is the description of abelian extensions of global fields and local. The label \ class eld refers to a eld extension satisfying a technical property that is historically related to ideal class groups, and one of the main theorems is that class elds are the same as abelian extensions. Generalized ideal class groups and the artin reciprocity law. Prime ideals of ok are called finite primes to distinguish them from infinite. Note if a2i then a2iby property ii, so the nonzero elements of ioccur in pairs a. The class group is a measure of the extent to which unique factorization fails in the ring of integers of k. Class field theory in characteristic p its origin and. Introduction class eld theory is the description of abelian extensions of global elds and local elds.

Since f0g 0 and z 1, we may assume z is a nonzero proper ideal. Some pari programs have bringed out a property for the nongenus part of the class number of imaginary quadratic fields of fixed signature, with respect to vd. He proved the fundamental theorems of abelian class field theory. In number theory, the ideal class group or class group of an algebraic number field k is the quotient group j k p k where j k is the group of fractional ideals of the ring of integers of k, and p k is its subgroup of principal ideals. History of class field theory 3 it is unrami ed over kand every ideal of kbecomes principal in it. The order of the group, which is finite, is called. Pdf on the principal ideal theorem in arithmetic topology. Intro to class field theory and the chebotarev theorem.

In the rst half, we prove preliminary results and state the main results. There is a generalization of the principal ideal theorem to ray class groups. This paper introduces basic theorems of class field theory and. This observation has a long history going back to fermat and euler. Let k be a number field, a ok, and p a prime ideal of k. Hilbert will include these properties as part of his general conjectures on hilbert class elds. Class field theory describes the abelian extensions of a local or global field in terms of the. He introduced an important new approach into algebraic number.