Related papers: Model Completeness for Henselian Fields with finit…

We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the residue field and value group. We address the case in which the valued field has a value group…

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are…

For a finite totally ramified extension $L$ of a complete discrete valuation field $K$ with the perfect residue field of characteristic $p>0$, it is known that $L/K$ is an abelian extension if the upper ramification breaks are integers and…

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…

We show that a field $K$ is model complete (in the language of rings) if and only if the Heisenberg group $H(K)$ is model complete (in the language of groups). To show that, we extend Levchuk's result about automorphisms of $H(K)$ to the…

A henselian valued field $K$ is called a tame field if its algebraic closure $\tilde{K}$ is a tame extension, that is, the ramification field of the normal extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal…

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more…

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure…

We prove that a valued field of positive characteristic $p$ that has only finitely many distinct Artin-Schreier extensions (which is a property of infinite NTP$_2$ fields) is dense in its perfect hull. As a consequence, it is a deeply…

Sen's theorem on the ramification of a $p$-adic analytic Galois extension of $p$-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing $p$-adic Hodge theory for…

The aim of this article is to give a self-contained account of the algebra and model theory of Cohen rings, a natural generalization of Witt rings. Witt rings are only valuation rings in case the residue field is perfect, and Cohen rings…

We conjecture that a $p$-algebra over a complete discrete valued field $K$ contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyclic maximal subfield. We prove the conjecture in several…

We give a characterization of finitely ramified $\omega$-pseudo complete valued fields of mixed characteristic $(0, p)$, with fixed residue field $k$ and value group $G$ of cardinality $\aleph_{1}$, in terms of a Hahn-like construction over…

We study the model theory of finitely ramified henselian valued fields of fixed initial ramification, obtaining versions of the Ax-Kochen-Ershov principle as follows. We identify the induced structure on the residue field and show that once…

We consider the class of complete discretely valued fields such that the residue field is of prime characteristic p and the cardinality of a $p$-base is 1. This class includes two-dimensional local and local-global fields. A new definition…

In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real…

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th…

Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$,…

In this note we study sets of NIP formulas in some theories of fields and valued fields, with a special focus on the sets of quantifier-free and existential formulas. First, we give a new proof of the fact that Separably Closed Valued…

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote…