Related papers: Model theory of fields with finite group scheme ac…
In this preliminary note we prove that the theory of valued fields equipped with an action of a given finite group has a model companion.
We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general…
We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.
In this article, we study several problems related to virtual traces for finite group actions on schemes of finite type over an algebraically closed field. We also discuss applications to fixed point sets. Our results generalize previous…
For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point space and for stable homotopy…
We survey some results on the structure of the groups which are definable in theories of fields involved in the applications of model theory to Diophantine geometry. We focus more particularly on separably closed fields of finite degree of…
Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a flat $R$-scheme of finite type and $G$ a finite flat group scheme acting on $X$ so that $G\_K$ is faithful on the generic fibre $X\_K$. We prove that there is an…
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…
Let G be a finite group. We explore the model theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential field automorphisms. In the language of…
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first order theory of finite groups. The focus is on concepts from stability theory…
We survey some results concerning finite group actions on products of spheres.
The author proposes a method for investigating actions of finite groups on aspherical spaces. Complete homotopy classification of free actions of finite groups on aspherical spaces is obtained. Also there are some results about non-free…
We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of certain orbifold fundamental groups.
We study field theory models in the context of a gravitational theory based on the requirement that the measure of integration in the action is not necessarily \sqrt{-g} but it is determined dynamically through additional degrees of…
We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.
In recent years the complex action problem of lattice field theory at finite density was overcome for several system by mapping them to dual variables (flux lines and surfaces). We illustrate this mapping for the case of the U(1) gauge…
We develop a Galois theory of commutative rings under actions of finite inverse semigroups. We present equivalences for the definition of Galois extension as well as a Galois correspondence theorem. We also show how the theory behaves in…
We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a…
McGrail has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe has shown the existence of a model completion for the univeral theory…
Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a…