scholarly journals The different and differentials of local fields with imperfect residue fields

1997 ◽  
Vol 40 (2) ◽  
pp. 353-365 ◽  
Author(s):  
Bart de Smit
Keyword(s):  
Galois Extension ◽  
Correction Term ◽  
Residue Field ◽  
Field Extension ◽  
Local Fields ◽  
Valuation Rings ◽  
Complete Field ◽  
Residue Fields ◽  
Residue Characteristic ◽  
Module Of Differentials

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

scholarly journals Faltings extension and Hodge-Tate filtration for abelian varieties over p-adic local fields with imperfect residue fields

2020 ◽  
pp. 1-17
Author(s):  
Tongmu He
Keyword(s):  
Abelian Varieties ◽  
Residue Field ◽  
Local Fields ◽  
Field Case ◽  
Residue Fields ◽  
Valuation Field

Abstract Let K be a complete discrete valuation field of characteristic $0$ , with not necessarily perfect residue field of characteristic $p>0$ . We define a Faltings extension of $\mathcal {O}_K$ over $\mathbb {Z}_p$ , and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.

scholarly journals Complexity of OM factorizations of polynomials over local fields

2013 ◽  
Vol 16 ◽  
pp. 139-171 ◽  
Author(s):  
Jens-Dietrich Bauch ◽  
Enric Nart ◽  
Hayden D. Stainsby
Keyword(s):  
Maximal Ideal ◽  
Valuation Ring ◽  
Residue Field ◽  
Local Fields ◽  
Single Factor ◽  
Locally Compact ◽  
Complete Field

AbstractLet $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $ \mathcal{O} $ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree $n$. Let $\delta = v(\mathrm{Disc} (F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, for a prescribed precision $\nu $. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O({n}^{2+ \epsilon } + {n}^{1+ \epsilon } {\delta }^{2+ \epsilon } + {n}^{2} {\nu }^{1+ \epsilon } )$ word operations for the complexity of the computation of a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, assuming that the residue field of $k$ is small.

Ramification Theory for Valuations of Arbitrary Rank

1974 ◽  
Vol 26 (4) ◽  
pp. 908-916 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Galois Extension ◽  
Residue Field ◽  
Field Extension ◽  
Ordered Group ◽  
Valued Fields ◽  
Arbitrary Rank ◽  
Ramification Theory

Throughout, we consider a finite Galois extension L|K of non-archimedian valued fields which are maximally complete [2, Chapter 2], Let v denote the valuation on L and let L* denote the group of non-zero elements of L. We mayidentify the value group v(L*) of L with a subgroup of D, where D denotes the minimal divisible ordered group containing v(K*). We denote the residue field of L by , and will always assume that the field extension is separable. The characteristic of will invariably be denoted by p ; much of what follows is trivial in case p = 0.

Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor

2000 ◽  
Vol 52 (6) ◽  
pp. 1269-1309 ◽  
Author(s):  
Luca Spriano
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Field Extension ◽  
Ring Extension ◽  
General Hypothesis ◽  
Valuation Rings ◽  
Ramification Theory

AbstractWe study extensions L/K of complete discrete valuation fields K with residue field of characteristic p > 0, which we do not assume to be perfect. Our work concerns ramification theory for such extensions, in particular we show that all classical properties which are true under the hypothesis “the residue field extensionis separable” are still valid under the more general hypothesis that the valuation ring extension is monogenic. We also show that conversely, if classical ramification properties hold true for an extension L/K, then the extension of valuation rings is monogenic. These are the “well ramified” extensions. We show that there are only three possible types of well ramified extensions and we give examples. In the last part of the paper we consider, for the three types, Kato’s generalization of the conductor, which we show how to bound in certain cases.

scholarly journals Integral Normal Bases in Galois Extensions of Local Fields

1970 ◽  
Vol 39 ◽  
pp. 141-148 ◽  
Author(s):  
S. Ullom
Keyword(s):  
Prime Ideal ◽  
Galois Group ◽  
Galois Extension ◽  
Residue Field ◽  
Local Fields ◽  
Galois Extensions ◽  
Normal Bases ◽  
Ring Of Integers

Throughout this paper F denotes a field complete with respect to a discrete valuation, kF the residue field of F, K/F a finite Galois extension with Galois group G = G(K/F). The ring of integers 0K of K contains the (unique) prime ideal ; the collection of ideals n for all integers n are ambiguous ideals i.e. G-modules.

Some model theory for almost real closed fields

1996 ◽  
Vol 61 (4) ◽  
pp. 1121-1152 ◽  
Author(s):  
Françoise Delon ◽  
Rafel Farré
Keyword(s):  
Model Theory ◽  
Residue Field ◽  
Elementary Equivalence ◽  
First Order ◽  
Valuation Rings ◽  
Closed Fields ◽  
Theory Of Fields ◽  
Model Theory Of Fields ◽  
Elementary Inclusion ◽  
Henselian Valuation

AbstractWe study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

Valued Differential Fields

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Asymptotic Behavior ◽  
Residue Field ◽  
Field Extension ◽  
Field Extensions ◽  
Differential Field ◽  
Differential Fields ◽  
Dominant Part

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous P ∈ K K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

scholarly journals On value groups and residue fields of some valued function fields

1994 ◽  
Vol 37 (3) ◽  
pp. 445-454
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Cyclic Group ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Torsion Group ◽  
Transcendence Degree ◽  
Manuscripta Math ◽  
Direct Sum ◽  
Residue Fields ◽  
The One

Let K = K0(x, y) be a function field of transcendence degree one over a field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a residue field k0 and υ be a prolongation of υ to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the residue fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).

scholarly journals Crossed Products and Ramification

1966 ◽  
Vol 28 ◽  
pp. 85-111 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Rank One ◽  
Definition Of

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

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