The Witt group of a discretely valued field and the fundamental filtration

2021 ◽  
Author(s):  
Jón Kr. Arason
Keyword(s):  
Witt Group ◽  
Valued Field

The Witt group of a discretely valued field

2018 ◽  
Vol 511 ◽  
pp. 102-113 ◽  
Author(s):  
Jón Kr. Arason
Keyword(s):  
Witt Group ◽  
Valued Field

Preliminaries

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Galois Coverings ◽  
Definable Type ◽  
Definable Sets ◽  
Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

Witt group

2017 ◽  
pp. 163-174
Author(s):  
Kazimierz Szymiczek
Keyword(s):  
Witt Group

The Witt group of a smooth complex surface

1987 ◽  
Vol 277 (3) ◽  
pp. 469-481 ◽  
Author(s):  
Fernando Fern�ndez-Carmena
Keyword(s):  
Complex Surface ◽  
Witt Group ◽  
Smooth Complex

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS

2008 ◽  
Vol 08 (01) ◽  
pp. 1-22 ◽  
Author(s):  
DEIRDRE HASKELL ◽  
YOAV YAFFE
Keyword(s):  
Valuation Ring ◽  
Uniform Method ◽  
Algebraic Characterization ◽  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Complete Theories ◽  
The Given ◽  
Relevant Class

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.

scholarly journals ON IRREDUCIBLE FACTORS OF POLYNOMIALS OVER COMPLETE FIELDS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250125 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
SANJEEV KUMAR
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Manuscripta Math ◽  
Algebraic Number Fields ◽  
Valued Fields ◽  
Valued Field ◽  
Transcendental Extension ◽  
Simple Transcendental Extension ◽  
Irreducibility Criterion ◽  
Hensel’S Lemma

Let (K, v) be a complete rank-1 valued field. In this paper, we extend classical Hensel's Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind's theorem regarding splitting of primes in algebraic number fields. We also deduce an irreducibility criterion for polynomials over rank-1 valued fields which extends already known generalizations of Schönemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in Khanduja and Khassa [Manuscripta Math.134(1–2) (2010) 215–224] is also obtained as a corollary of the main result.

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