Related papers: The algebraic theory of valued fields

We give a proposal for future development of the model theory of valued fields. We also summarize some recent results on p-adic numbers.

In this article we further develop the theory of valuation independence and study its relation with classical notions in valuation theory such as immediate and defectless extensions. We use this general theory to settle two open questions…

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…

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an…

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups,…

We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow…

We give a survey on recent developments in the model theory of valued fields since the introduction of the notion of ``tame valued field'', and of the modifications and generalizations of this notion.

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…

We develop an extension of valuations theorem for suitable extensions of idempotent semirings. As an application, we give a new proof for the classical case of fields. Along the way, we develop characteristic one analogues of some central…

In this paper, we study extensions of valuations over algebraic field extensions without the use of the Axiom of Choice. We show a bijection between the extensions of a valuation and the maximal ideals of the relative integral closure of…

Let D be a valued division algebra, finite-dimensional over its center F. Assume D has an unramified splitting field. The paper shows that if D contains a maximal subfield which is Galois over F (i.e. D is a crossed product) then the…

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…

We study the relation between two important classes of valued fields: tame fields and defectless fields. We show that in the case of valued fields of equal characteristic or rank one valued fields of mixed characteristic, tame fields are…

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…

We study in detail the valuation theory of deeply ramified fields and introduce and investigate several other related classes of valued fields. Further, a classification of defect extensions of prime degree of valued fields that was earlier…

Fifteen years after their discovery, ample fields now stand at the center of research in contemporary Galois theory and attract more and more attention also from other areas of mathematics. This survey gives an introduction to the theory of…

We compute the Galois cohomology of any $p$-adic valuation field extension of a pre-perfectoid field. Moreover, we obtain a generalization and also a new proof of the classical results of Tate and Hyodo on discrete valuation fields, without…

Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.

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…

One of the fundamental questions in current field theory, related to Grothendieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the…