Fields with two incomparable henselian valuation rings

1978 ◽  
Vol 23 (4) ◽  
pp. 373-385 ◽  
Author(s):  
Antonio J. Engler
Keyword(s):  
Valuation Rings ◽  
Henselian Valuation

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.

scholarly journals DEFINABLE HENSELIAN VALUATION RINGS

2015 ◽  
Vol 80 (4) ◽  
pp. 1260-1267 ◽  
Author(s):  
ALEXANDER PRESTEL
Keyword(s):  
Number Field ◽  
Function Field ◽  
Residue Class ◽  
Elementary Theory ◽  
Complex Curve ◽  
Valuation Rings ◽  
Henselian Valued Fields ◽  
Residue Class Field ◽  
Image Position ◽  
Henselian Valuation

AbstractWe give model theoretic criteria for the existence of ∃∀ and ∀∃- formulas in the ring language to define uniformly the valuation rings ${\cal O}$ of models $\left( {K,\,{\cal O}} \right)$ of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings ${\cal O}$ of valued henselian fields $\left( {K,\,{\cal O}} \right)$ whose residue class field k is finite, pseudofinite, or hilbertian. We also obtain ∀∃-formulas φ2 and φ4 such that φ2 defines uniformly k[[t]] in k(t) whenever k is finite or the function field of a real or complex curve, and φ4 replaces φ2 if k is any number field.

Fields with henselian valuation rings

1977 ◽  
Vol 152 (2) ◽  
pp. 191-193 ◽  
Author(s):  
Otto Endler ◽  
Antonio Jos� Engler
Keyword(s):  
Valuation Rings ◽  
Henselian Valuation

scholarly journals EXISTENTIAL ∅-DEFINABILITY OF HENSELIAN VALUATION RINGS

2015 ◽  
Vol 80 (1) ◽  
pp. 301-307 ◽  
Author(s):  
ARNO FEHM
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Quotient Field ◽  
Valued Fields ◽  
Valuation Rings ◽  
Henselian Valued Fields ◽  
Residue Fields ◽  
Definition Of ◽  
Formal Power ◽  
Henselian Valuation

AbstractIn [1], Anscombe and Koenigsmann give an existential ∅-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 results for henselian valued fields with finite or pseudo-algebraically closed residue fields.

scholarly journals On the Theory of Henselian Rings, II

1954 ◽  
Vol 7 ◽  
pp. 1-19 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
General Theory ◽  
Local Ring ◽  
Maximal Ideal ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Henselian Valuation

In a previous paper, we studied a general theory of integrally closed Henselian integrity domains and some properties of Henselian valuation rings. The present paper is its continuation. The main aim of the present paper is to study a general theory of Henselian local integrity domains in the present paper we call a ring o a local ring if o is a quasi-local ring and if the intersection of all powers of the maximal ideal of o is zero, and in this case we introduce a topology by taking the system of all powers of the maximal ideal as a system of neighbourhoods of zero.

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.

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.

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