相关论文: Galois theory, motives and transcendental numbers

We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the…

Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…

In the present paper, we give a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational function K(x), where K is a finite extension of a field of rational functions k(q), with…

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field:…

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to…

In this article, we realize some groups as Galois groups over rational numbers and finite extension of rational numbers by studying right splitting of some exact sequences, Galois correspondence and algebraic operations on Galois…

Let k be a base field of positive characteristic. Making use of topological periodic cyclic homology, we start by proving that the category of noncommutative numerical motives over k is abelian semi-simple, as conjectured by Kontsevich.…

We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…

We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of…

A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field…

The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types…

In this exposition we discuss the theory of algebraic extensions of valued fields. Our approach is mostly through Galois theory. Most of the results are well-known, but some are new. No previous knowledge on the theory of valuations is…

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to…

We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the…

Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of…

We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference…

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…

Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a…

This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for…