Small generators of the ideal class group

Small generators of the ideal class group

Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group C K of a number field K can be generated by the prime ideals of K having norm smaller than 12 ( log |Discriminant(K)| )2 . This result is essential for the computation of the class group and units of K by Buchmann’s algorithm, currently the fastest known. However, once C K has been computed, one notices that ...

The Ideal Class Group

We present a concise and self-contained definition of the ideal class group, which is useful for proving facts about zero sets of Diophantine equations, and discuss a few relevant key facts. We approach this by first assembling some preliminary definitions regarding algebraic integers, and subsequently delving into several useful results about lattices, including Minkowski’s lemma. Then, return...

The Ideal Class Group

When we form a finite algebraic extension of Q, we are not guaranteed that the ring of integers, O, in our extension will be a unique factorization domain (UFD). We can obtain a measure of how far O is from being a UFD by computing the class number which is defined as the order of the ideal class group. This paper describes the ideal class group and provides examples of how to compute this grou...

Ideal Class Group Annihilators

A Jesús Muñoz Díaz por su cumpleaños Contents 1. Introduction. 1 2. Elliptic sheaves and Hecke correspondences. Euler products 3 2.1. Elliptic sheaves 3 2.2. ∞-Level structures 5 2.3. Hecke correspondences 6 2.4. Euler products 8 3. Isogenies and Hecke correspondences 9 3.1. Isogenies for elliptic sheaves 9 3.2. Trivial correspondences 11 4. Ideal class group anhilators for the cyclotomic funct...

The Generators of the 2-class Group of Some Fields