Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials

2015 ◽  
Vol 58 (2) ◽  
pp. 225-232
Author(s):  
Kamal Aghigh ◽  
Azadeh Nikseresht
Keyword(s):  
Algebraic Closure ◽  
Unique Extension ◽  
Irreducible Polynomials ◽  
Henselian Valuation

AbstractLet v be a henselian valuation of any rank of a field K and let be the unique extension of v to a fixed algebraic closure of K. In 2005, we studied properties of those pairs (θ,α) of elements of with where α is an element of smallest degree over K such thatSuch pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.

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

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.

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.

INVARIANTS OF DEFECTLESS IRREDUCIBLE POLYNOMIALS

2010 ◽  
Vol 09 (04) ◽  
pp. 603-631 ◽  
Author(s):  
RON BROWN ◽  
JONATHAN L. MERZEL
Keyword(s):  
Irreducible Polynomial ◽  
Newton Polygon ◽  
Extension Field ◽  
Irreducible Polynomials ◽  
Technical Result ◽  
Valued Field ◽  
Minimal Pairs ◽  
Simple Characterization ◽  
Polynomial Extensions

Defectless irreducible polynomials over a Henselian valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete (also called "saturated") distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case in which a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f. In this case a (best possible) result is given showing that a sufficiently good approximation in an extension field K of F to a root of a defectless polynomial f over F guarantees the existence of an exact root of f in K. Also in the tame case a (best possible) result is given describing when a polynomial is sufficiently close to a defectless polynomial so as to guarantee that the roots of the two polynomials generate the same extension fields. Another application in the tame case gives a simple characterization of the minimal pairs (in the sense of N. Popescu et al.). A key technical result is a computation in the tame case of the Newton polygon of f(x+α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja's characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials. Much of the work described here will be seen not to require the hypothesis that (F, v) is Henselian.

Another Uniqueness Proof of the Dickson Simple Group G2(3)

2011 ◽  
Vol 18 (02) ◽  
pp. 181-210
Author(s):  
Gerhard O. Michler ◽  
Lizhong Wang
Keyword(s):  
Simple Group ◽  
Cyclic Group ◽  
Unique Extension ◽  
Uniqueness Criterion ◽  
Central Product ◽  
Uniqueness Proof

In this article, we give a self-contained uniqueness proof for the Dickson simple group G=G2(3) using the first author's uniqueness criterion. The uniqueness proof for G2(3) was first given by Janko. His proof depends on Thompson's deep and technical characterization of G2(3). Let H′ be the amalgamated central product of SL 2(3) with itself. Then there is a unique extension H of H′ by a cyclic group of order 2 such that H has a center of order 2 and both factors SL 2(3) are normal in H. We prove that any simple group G having a 2-central involution z with centralizer CG(z)≅ H is isomorphic to G2(3).

Small theories of Boolean ordered o-minimal structures

2002 ◽  
Vol 67 (4) ◽  
pp. 1385-1390 ◽  
Author(s):  
Roman Wencel
Keyword(s):  
Algebraic Closure ◽  
Complete Characterization ◽  
Finite Sets

AbstractWe investigate small theories of Boolean ordered o-minimal structures. We prove that such theories are ℵ0-categorical. We give a complete characterization of their models up to bi-interpretability of the language. We investigate types over finite sets, formulas and the notions of definable and algebraic closure.

A note on subgroups of the automorphism group of a saturated model, and regular types

1989 ◽  
Vol 54 (3) ◽  
pp. 858-864 ◽  
Author(s):  
A. Pillay
Keyword(s):  
Automorphism Group ◽  
Algebraic Closure ◽  
Saturated Model ◽  
Superstable Theory ◽  
Alternative Characterization ◽  
Finite Set ◽  
Superstable Theories

AbstractLet M be a saturated model of a superstable theory and let G = Aut(M). We study subgroups H of G which contain G(A), A the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types p in the context of p-simple types.

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.

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