Related papers: Open Problems in the Theory of Ample Fields

This note describes some open problems that can be examined with the purpose of gaining additional insight of how to solve the problem of finding a general classification of geodetic graphs

In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix…

We discuss some open problems in a program of constructing and studying two-dimensional conformal field theories using the representation theory of vertex operator algebras.

Glasses at low temperature fluctuate around their inherent states; glassy anomalies reflect the structure of these states. Recently there have been numerous observations of long-range stress correlations in glassy materials, from…

We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to 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…

The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic…

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois…

The notion of `Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had new applications. In this work we develop a…

Effective field theories exploit a separation of scales in physical systems in order to perform systematically improvable, model-independent calculations. They are ideally suited to describe universal aspects of a wide range of physical…

This report records a large number of open problems in Affine Algebraic Geometry that were proposed by participants in a Conference on Open Algebraic Varieties at the Centre de Recherches en Mathematiques in Montreal at December 1994.

We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois…

Let L be a Galois extension of a countable Hilbertian field K. Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L/K are.

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…

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and…

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends…

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from…

The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also…