Related papers: Notes on Nonlinear Number Fields

We give a new construction of nonlinear error-correcting codes over suitable finite fields k from the geometry of modular curves with many rational points over k, combining two recent improvements on Goppa's construction. The resulting…

The article investigates several questions about field generated by certain formal group laws. This is continuation of a study initiated by the author in prior article titled 'Field Generated by Division Points of Certain Formal Group…

A diagram approach to classical nonlinear stochastic field theory is introduced. This approach is intended to serve as a link between quantum and classical field theories, resulting in an independent constructive characterisation of the…

This paper extends Hopf-Galois theory to infinite field extensions and provides a natural definition of subextensions. For separable (possibly infinite) Hopf-Galois extensions, it provides a Galois correspondence. This correspondence also…

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an…

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives…

We study abelian subgroups of Galois groups of function fields.

This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of finite group schemes with techniques coming from relative algebraic…

We give a nonstandard treatment of the notion of ends of proper geodesic metric spaces. We then apply this nonstandard treatment to Cayley graphs of finitely generated groups and give nonstandard proofs of many of the fundamental results…

Generalized non-autonomous linear celullar automata are systems of linear difference equations with many variables that can be seen as convolution equations in a discrete group. We study those systems from the stand point of the Galois…

Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.

We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this…

The lecture notes below correspond to the course given by the author in occasion of the VIASM school on Number Theory (18-24 June 2018, Hanoi). We have chosen to omit the proofs that are already presented in details in many references in…

We develop here the algebra of the differential field of transseries and of related valued differential fields. This book contains in particular our recently obtained decisive positive results on the model theory of these structures.

We survey some results relating noncommutative geometry to the class field theory of number fields. These results appear within the context of quantum statistical mechanics where some arithmetic properties of a given number field can be…

We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic…

We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…

A new approach to constructing the noncommutative scalar field theory is presented. Not only between x_i and p_j, we impose commutation relations between x_is as well as p_js, and give a new representation of x_i,p_js. We carry out both…

The purpose of this note is twofold. First, we survey results on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.

A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of…