Related papers: Computing class groups by induction with generalis…

For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the…

For each finite subgroup $G$ of $PGL_2(\mathbb{Q})$, and for each integer $n$ coprime to $6$, we construct explicitly infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ and whose ideal class group has $n$-rank at least…

A polynomial time algorithm to give a complete description of all subfields of a given number field was given in an article by van Hoeij et al. This article reports on a massive speedup of this algorithm. This is primary achieved by our new…

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…

In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity…

This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for…

We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The…

We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k$-algebra $A$, we consider the category of all connected gradings of $A$ by a group $G$ and we…

We present a family of algorithms for computing the Galois group of a polynomial defined over a $p$-adic field. Apart from the "naive" algorithm, these are the first general algorithms for this task. As an application, we compute the Galois…

Let L be an abelian number field of degree n with Galois group G. In this paper we study how to compute efficiently a normal integral basis for L, if there is at least one, assuming that the group G and an integral basis for L are known.

Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a…

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division…

Given a Mersenne prime $q$ and a positive even integer $e$, let $F$ and $E$ be the fields of orders $q$ and $q^e$ respectively. Let $C$ be a cyclic subgroup of $E^\times$ whose index in $E^\times$ is divisible only by primes dividing $q -…

In this article we show how Gr\"un's results in group theory can be used for studying the structure of class groups in normal extensions.

Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation…

We construct an infinite family of real cyclotomic fields with non-trivial class group. This result generalizes the result in [1] in the sense that our family includes theirs.

Counting number fields with prescribed Galois group is an enduring challenge in arithmetic statistics. Using the determinant method, we provide an upper bound for even groups, which is new in some cases.

It is well known that the Galois group of an extension puts constraints on the structure of the relative ideal class groups. Using only basic parts of the theory of group representations, we give a unified approach to such results.