Related papers: Growth Results and Euclidean Ideals

Lenstra introduced the notion of a Euclidean ideal class, which is a generalization of the Euclidean domain. Lenstra also proved that the Euclidean ideal in a number field $K$ implies that the class group of $K$ is cyclic. We construct a…

Lenstra introduced the notion of the Euclidean ideal class, a generalization of the Euclidean domain that captures cyclic class groups. In this article, we establish the existence of Euclidean ideal classes in abelian quartic fields. As a…

We study Euclidean ideal classes in real biquadratic fields and obtain unconditional existence results via genus theory. Lenstra showed (assuming the Generalized Riemann Hypothesis) that a number field with unit rank at least one admits a…

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the…

H. W. Lenstra \cite{lenstra} introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal…

In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean…

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…

We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. We also give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class…

We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.

An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…

This article focuses on some rings of integers of number fields which are known to be norm-Euclidean domains, but for which no explicit algorithm computing the Euclidean division has yet been studied or implemented. The rings of integers we…

The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…

We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating of computational complexity of algorithms which generates special primes. Such primes are needed to…

We describe a new algorithm for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve…

We introduce axiomatically the ring $\bf{Z}_\kappa$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $\kappa$-sequences…

In [BDyDF08] Belabas, Diaz y Diaz and Friedman show a way to determine, assuming the Generalized Riemann Hypothesis, a set of prime ideals that generate the class group of a number field. Their method is efficient because it produces a set…

In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals…

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…

It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic…

The P\'{o}lya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large…