On the vector space defect for valued field extensions

2000 ◽  
pp. 209-217
Author(s):  
Jack Ohm
Keyword(s):  
Vector Space ◽  
Field Extensions ◽  
Valued Field

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.

scholarly journals Counting the number of distinct distances of elements in valued field extensions

2018 ◽  
Vol 509 ◽  
pp. 192-211 ◽  
Author(s):  
Anna Blaszczok ◽  
Franz-Viktor Kuhlmann
Keyword(s):  
Field Extensions ◽  
Valued Field

scholarly journals A p-adic Montel Theorem and locally polynomial functions

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 159-166
Author(s):  
J.M. Almira ◽  
Kh.F. Abu-Helaiel
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Vector Space ◽  
Normed Space ◽  
Characteristic Zero ◽  
Continuous Functions ◽  
Polynomial Functions ◽  
Valued Field ◽  
Q Vector

We prove a version of Montel?s Theorem for the case of continuous functions defined over the field Qp of p-adic numbers. In particular, we prove that, if ?m+1 h0 f (x) = 0 for all x ? Qp, and h0 satisfies |h0|p = p?N0, then, for all x0 ? Qp, the restriction of f over the set x0 + pN0Zp coincides with a polynomial px0 (x) = a0(x0) + a1(x0)x +...+ am(x0)xm. Motivated by this result, we compute the general solution of the functional equation with restrictions given by ?m+1 h f (x) = 0 (x ? X and h ? BX(r) = {x ? X : ?x? ? r}), whenever f : X ? Y, X is an ultrametric normed space over a non-Archimedean valued field (K, |?|) of characteristic zero, and Y is a Q-vector space. By obvious reasons, we call these functions uniformly locally polynomial.

scholarly journals Distances of elements in valued field extensions

2018 ◽  
Vol 159 (3-4) ◽  
pp. 397-429
Author(s):  
Anna Blaszczok
Keyword(s):  
Field Extensions ◽  
Valued Field

scholarly journals On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

2021 ◽  
Author(s):  
Lhoussain El Fadil ◽  
Mohamed Faris
Keyword(s):  
Algebraic Number Theory ◽  
Sufficient Condition ◽  
Polynomial Factorization ◽  
General Version ◽  
Field Extensions ◽  
Valued Field ◽  
Rank One ◽  
Key Polynomials ◽  
Irreducibility Criterion

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L=Kα, we give a method to describe all absolute values of L extending ∣∣, where K is a discrete rank one valued field.

scholarly journals Graded transcendental extensions of graded fields

2003 ◽  
Vol 2003 (70) ◽  
pp. 4435-4446
Author(s):  
M. Boulagouaz
Keyword(s):  
Field Extension ◽  
Field Extensions ◽  
Valued Field

We study transcendency properties for graded field extension and give an application to valued field extensions.

scholarly journals Identities for field extensions generalizing the Ohno–Nakagawa relations

2015 ◽  
Vol 151 (11) ◽  
pp. 2059-2075 ◽  
Author(s):  
Henri Cohen ◽  
Simon Rubinstein-Salzedo ◽  
Frank Thorne
Keyword(s):  
Vector Space ◽  
Functional Equations ◽  
Zeta Functions ◽  
Reflection Principle ◽  
Galois Groups ◽  
Field Extensions ◽  
Prehomogeneous Vector Space ◽  
Counting Functions ◽  
Cubic Fields ◽  
Cubic Forms

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

Field Extensions and Galois Theory

1984 ◽  
Author(s):  
Julio R. Bastida ◽  
Roger Lyndon
Keyword(s):  
Galois Theory ◽  
Field Extensions

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.

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