scholarly journals Definable V-topologies, Henselianity and NIP

2019 ◽  
Vol 20 (02) ◽  
pp. 2050008
Author(s):  
Yatir Halevi ◽  
Assaf Hasson ◽  
Franziska Jahnke
Keyword(s):  
Finite Field ◽  
Algebraic Closure ◽  
Residue Field ◽  
Henselian Valuation

We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian.

Constructing complete distinguished chains with given invariants

2014 ◽  
Vol 14 (03) ◽  
pp. 1550026
Author(s):  
Kamal Aghigh ◽  
Azadeh Nikseresht
Keyword(s):  
Algebraic Closure ◽  
Residue Field ◽  
Finite Sequence ◽  
Unique Extension ◽  
Arbitrary Rank ◽  
A Chain ◽  
Henselian Valuation

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and [Formula: see text] be the unique extension of v to a fixed algebraic closure [Formula: see text] of K with value group [Formula: see text]. It is known that a complete distinguished chain for an element θ belonging to [Formula: see text] with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of [Formula: see text], a tower of fields, together with a sequence of elements belonging to [Formula: see text] which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of [Formula: see text] containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of [Formula: see text] satisfying certain properties, it is shown that there exists a complete distinguished chain for an element [Formula: see text] associated to these invariants. We use the notion of lifting of polynomials to construct it.

scholarly journals Global Norm-Residue Map over Quasi-Finite Field

1966 ◽  
Vol 27 (1) ◽  
pp. 323-329 ◽  
Author(s):  
D. S. Rim ◽  
G. Whaples
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Local Field ◽  
Algebraic Closure ◽  
Residue Field ◽  
Additive Group ◽  
Local Fields ◽  
Ordinary Sense ◽  
Reciprocity Law ◽  
Norm Residue

A field k is called quasi-finite if it is perfect and if Gk≈Ż where Gk is the Galois group of the algebraic closure kc over k and Ż is the completion of the additive group of the rational integers. The classical reciprocity law on the local field with finite residue field is well-known to hold on local fields with quasi-finite residue field ([4] [5]). Thus it is natural to ask if the global reciprocity law should hold in the ordinary sense (see § 1 below) on the function-fields of one variable over quasi-finite field. We consider here two basic prototypes of non-finite quasi-finite fields:

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

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 GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

2016 ◽  
Vol 224 (1) ◽  
pp. 93-167 ◽  
Author(s):  
JAY TAYLOR
Keyword(s):  
Finite Field ◽  
Wave Front ◽  
Irreducible Character ◽  
Prime Order ◽  
Algebraic Closure ◽  
Characteristic Functions ◽  
Rational Structure ◽  
Reductive Algebraic Group ◽  
Wave Front Set ◽  
Frobenius Endomorphism

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

scholarly journals Character sheaves and generalized Springer correspondence

2003 ◽  
Vol 170 ◽  
pp. 47-72 ◽  
Author(s):  
Anne-Marie Aubert
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Closure ◽  
Reductive Algebraic Group ◽  
Good For ◽  
Character Sheaves ◽  
Springer Correspondence

AbstractLetGbe a connected reductive algebraic group over an algebraic closure of a finite field of characteristicp. Under the assumption thatpis good forG, we prove that for each character sheafAonGwhich has nonzero restriction to the unipotent variety ofG, there exists a unipotent classCAcanonically attached toA, such thatAhas non-zero restriction onCA, and any unipotent classCinGon whichAhas non-zero restriction has dimension strictly smaller than that ofCA.

Counting points of given height that generate a quadratic extension of a function field

2015 ◽  
Vol 11 (02) ◽  
pp. 569-592 ◽  
Author(s):  
David Kettlestrings ◽  
Jeffrey Lin Thunder
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Function Field ◽  
Asymptotic Estimate ◽  
Algebraic Closure ◽  
Quadratic Extension ◽  
Rational Functions ◽  
Rational Numbers ◽  
Algebraic Extension ◽  
Counting Points

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

The classification of small weakly minimal sets. III: Modules

1988 ◽  
Vol 53 (3) ◽  
pp. 975-979 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Finite Field ◽  
Algebraic Closure ◽  
Morley Rank ◽  
Minimal Sets

AbstractTheorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let A = acl(∅) ⋂ M and I = {r ∈ R: rM ⊂ A}. Notice that I is an ideal.(i) F = R/Iis a finite field.(ii) Suppose that a, b0,…,bn, ∈ M and . Then there are s, ri ∈ R, i ≤ n, such that sa + Σi≤nribi ∈ A and s ∉ I.It follows from Theorem A that algebraic closure in M is modular. Using this and results in [B1] and [B2], we obtainTheorem B. Let M be as in Theorem A. Then Vaught's conjecture holds for Th(M).

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

scholarly journals The factorable core of polynomials over finite fields

1990 ◽  
Vol 49 (2) ◽  
pp. 309-318 ◽  
Author(s):  
S. D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Permutation Polynomials ◽  
Linearized Polynomial ◽  
Polynomials Over Finite Fields ◽  
Value Sets

AbstractFor a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

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