A Note on Henselian Valuation Rings

1968 ◽  
Vol 11 (2) ◽  
pp. 185-189 ◽  
Author(s):  
Otto Endler
Keyword(s):  
Valuation Ring ◽  
Algebraic Closure ◽  
Valuation Rings ◽  
Rank One ◽  
Hensel’S Lemma ◽  
Henselian Valuation

Let K be a field and Ka its algebraic closure. A valuation a ring A of K is called henselian, if there is only one valuation ring C of Ka which lies over A (i.e. such that C ∩ K = A) or, equivalently, if Hensel's Lemma is valid for K, A (see [5], F). In the following, we shall consider only rank one valuation rings.

I-COMPACT RIGHT CHAIN DOMAINS

2011 ◽  
Vol 10 (06) ◽  
pp. 1107-1139
Author(s):  
H. H. BRUNGS ◽  
G. TÖRNER
Keyword(s):  
Valuation Ring ◽  
Convergent Sequence ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Real Numbers ◽  
Minimal Set ◽  
Valuation Rings ◽  
Rank One ◽  
Special Case ◽  
Convergent Sequences

Generalizing the concept of convergency to valued fields, Ostrowski in the 1930s introduced pseudo-convergent sequences. In the present paper we classify pseudo-convergent sequences in right chain domains R according to the prime ideal P associated to the breadth I of the sequence using an ideal theory developed for right cones in groups. The ring R is I-compact if every pseudo-convergent sequence in R with breadth I has a limit in R, and we construct right chain domains R which are I-compact only for right ideals I in particular subsets [Formula: see text] of the set of all right ideals of R. Krull's perfect valuation rings and then Ribenboim's notion of a valuation ring complete par étages, where [Formula: see text] is the minimal set containing the completely prime ideals in a commutative valuation ring, is a special case. For a non-discrete right invariant rank-one right chain domain R there are exactly two possibilities for the set [Formula: see text] if the value group of R is the group of real numbers under addition, and there are infinitely many possibilities for [Formula: see text] in all other cases.

On Chains Associated with Elements Algebraic over a Henselian Valued Field

2005 ◽  
Vol 12 (04) ◽  
pp. 607-616 ◽  
Author(s):  
Kamal Aghigh ◽  
Sudesh K. Khanduja
Keyword(s):  
Algebraic Closure ◽  
Minimal Degree ◽  
Unique Extension ◽  
Arbitrary Rank ◽  
Valued Field ◽  
Rank One ◽  
A Chain ◽  
Henselian Valuation

Let v be a henselian valuation of a field K, and [Formula: see text] be the (unique) extension of v to a fixed algebraic closure [Formula: see text] of K. For an element [Formula: see text], a chain [Formula: see text] of elements of [Formula: see text] such that θi is of minimal degree over K with the property that [Formula: see text] and θm ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each [Formula: see text] when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements [Formula: see text] for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ.

scholarly journals Ramification Theory for Extensions of Degree p. II

1972 ◽  
Vol 46 ◽  
pp. 97-109
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Root Of Unity ◽  
Ramification Index ◽  
Ramification Theory ◽  
Rank One ◽  
The Absolute

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

scholarly journals On three-variable expanders over finite valuation rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Le Quang Ham ◽  
Nguyen Van The ◽  
Phuc D. Tran ◽  
Le Anh Vinh
Keyword(s):  
Valuation Ring ◽  
Product Type ◽  
Quadratic Polynomial ◽  
Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

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.

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 A BETTER COMPARISON OF - AND -COHOMOLOGIES

2019 ◽  
Vol 236 ◽  
pp. 183-213
Author(s):  
SHANE KELLY
Keyword(s):  
Finite Rank ◽  
Valuation Ring ◽  
Direct Proof ◽  
Integral Closure ◽  
Motivic Cohomology ◽  
Valuation Rings

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

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.

Defining Families for Integral Domains of Real Finite Character

1972 ◽  
Vol 24 (6) ◽  
pp. 1170-1177 ◽  
Author(s):  
William Heinzer ◽  
Jack Ohm
Keyword(s):  
Zero Element ◽  
Finiteness Condition ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Rank One

Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).

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