The calculation of a large cubic class number with an application to real cyclotomic fields

The Calculation of a Large Cubic Class Number With an Application to Real Cyclotomic Fields

The class number of the cubic subfield of the pth cyclotomic field is calculated for the prime p = 11290018777. This is used to construct an example where the class number of the pth real cyclotomic field is larger than p. In a recent paper [1], it is shown that if the Generalized Riemann Hypothesis holds, then hp > p for the prime p = 11290018777. Here hp is the class number of Q(cos(27r/p)), ...

Quintic Polynomials and Real Cyclotomic Fields with Large Class Numbers

We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime p = 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an a...

Class-number Problems for Cubic Number Fields

On divisibility of the class number h+ of the real cyclotomic fields of prime degree l

In this paper, criteria of divisibility of the class number h+ of the real cyclotomic field Q(ζp +ζ−1 p ) of a prime conductor p and of a prime degree l by primes q the order modulo l of which is l−1 2 , are given. A corollary of these criteria is the possibility to make a computational proof that a given q does not divide h+ for any p (conductor) such that both p−1 2 , p−3 4 are primes. Note t...

Class number approximation in cubic function fields

A central problem in number theory and algebraic geometry is the determination of the size of the group of rational points on the Jacobian of an algebraic curve over a finite field. This question also has applications to cryptography, since cryptographic systems based on algebraic curves generally require a Jacobian of non-smooth order in order to foil certain types of attacks. There a variety ...