scholarly journals A definable Henselian valuation with high quantifier complexity

2015 ◽  
Vol 61 (4-5) ◽  
pp. 362-366
Author(s):  
Immanuel Halupczok ◽  
Franziska Jahnke
Keyword(s):  
Quantifier Complexity ◽  
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 ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD

2002 ◽  
Vol 45 (1) ◽  
pp. 219-227 ◽  
Author(s):  
Kamal Aghigh ◽  
Sudesh K. Khanduja
Keyword(s):  
Upper Bound ◽  
Algebraic Closure ◽  
Subject Classification ◽  
Unique Extension ◽  
Valued Fields ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
Alpha Beta ◽  
Henselian Valuation

AbstractLet $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

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

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.

scholarly journals DEFINABLE HENSELIAN VALUATIONS

2015 ◽  
Vol 80 (1) ◽  
pp. 85-99 ◽  
Author(s):  
FRANZISKA JAHNKE ◽  
JOCHEN KOENIGSMANN
Keyword(s):  
Galois Group ◽  
Residue Field ◽  
Absolute Galois Group ◽  
Valued Fields ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
The Absolute ◽  
Henselian Valuation

AbstractIn this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

XVIIème problème de Hilbert sur les corps chaîne-clos

1991 ◽  
Vol 56 (3) ◽  
pp. 853-861
Author(s):  
Françoise Delon et Danielle Gondard
Keyword(s):  
Residue Field ◽  
Algebraic Extension ◽  
Real Field ◽  
Closed Field ◽  
Several Variables ◽  
A Chain ◽  
Henselian Valuation

AbstractA chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, ΣK2n+2 ≠ ΣK2n) which does not admit any “faithful” algebraic extension, and can also be seen as a field having a Henselian valuation ν such that the residue field K/ν is real closed and the value group νK is odd divisible with ∣νK/2νK∣ = 2. If K admits only one such valuation, we show that f ∈ K(X) is in ΣK(X)2n for any real algebraic extension L of K,“f(L) ⊆ ΣL2n” holds. The conclusion is also true for K = R((t))(a chainable but not chain-closed field), and in the case n = 1 it holds for several variables and any real field K.

scholarly journals On the quantifier complexity of definable canonical Henselian valuations

2015 ◽  
Vol 61 (4-5) ◽  
pp. 347-361 ◽  
Author(s):  
Arno Fehm ◽  
Franziska Jahnke
Keyword(s):  
Quantifier Complexity
Sign in / Sign up

Export Citation Format

Share Document