Related papers: Formal prime ideals of infinite value and their al…

In this paper we study the rank one discrete valuations of the field $k((X_1,..., X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such…

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of…

Recently, Anscombe and Koenigsmann gave an existential 0-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability…

Non-archimedean fields with restricted analytic functions may not support a full exponential function, but they always have partial exponentials defined in convex subrings. On face of this, we study the first order theory of the class of…

In this paper we study the rank one discrete valuations of $k((X_1,... ,X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal $(\X)$. In sections 2 to 6 we give a construction of a system of parametric equations describing such…

Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $\mathcal{O}$ is the integral closure…

We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a…

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined…

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then…

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of $H^t_I(R)$ is finite for any ideal $I$ and for any $t \ge 0$…

In this paper we discuss stable forms of extensions of algebraic local rings along a valuation in all dimensions over a field k of characteristic zero, and generalize a formula of Ghezzi, H\`a and Kashcheyeva describing the extension of…

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…

Suppose $F$ is a field with valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study quasi-valuations on $E$ that extend $v$; in particular, their corresponding…

Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian domains whose fraction field is $K$, showing for example, that $R$ is excellent if and only if the Frobenius map is finite on $R$.…

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give criteria for a set to be…

In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{\nu^*}(S)$, is a finite ${\rm gr}_{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and…

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice…

In [Frobenius1896] it was shown that many important properties of a finite group could be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation,…

Let $k$ be a number field, $f(x)\in k[x]$ a polynomial over $k$ with $f(0)\neq 0$, and $\O_{k,S}^*$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural…