Related papers: Enumerative Galois theory for number fields
We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…
We enhance the analogy between field extensions and covering spaces by introducing the concept of splitting covering which correspondences to the splitting field in Galois theory. We define semi-topological Galois groups for Weierstrass…
A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this…
We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant…
Originally motivated by algebraic invariant theory, we present an algorithm to enumerate integer vectors modulo the action of a permutation group. This problem generalizes the generation of unlabeled graph up to an isomorphism. In this…
As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic…
For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the…
We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…
We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.
We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a…
We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…
We study the determinant of certain etale sheaves constructed via middle convolution in order to realize special linear groups regularly as Galois groups over the rationals.
A self-contained exposition is given of the topological and Galois-theoretic properties of the category of combinatorial 1-complexes, or graphs, very much in the spirit of Stallings. A number of classical, as well as some new results about…
The Galois group of a Schubert problem encodes some structure of its set of solutions. Galois groups are known for a few infinite families and some special problems, but what permutation groups may appear as a Galois group of a Schubert…
We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by…
In this article we study the Galois group of field generated by division points of special class of formal group laws and prove an equivalent condition for the group to be abelian. Further, we explore relations between the endomorphism ring…
The Galois group of a parameterized polynomial system of equations encodes the structure of the solutions. This monodromy group acts on the set of solutions for a general set of parameters, that is, on the fiber of a projection from the…
The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of GL(2,Z/nZ) where n is an integer. We know that, in general,…