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.

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

Bounds on Betti Numbers

1982 ◽  
Vol 34 (3) ◽  
pp. 589-592
Author(s):  
Mark Ramras
Keyword(s):  
Natural Number ◽  
Local Ring ◽  
Residue Class ◽  
Asymptotic Properties ◽  
Betti Numbers ◽  
Commutative Local Ring ◽  
Artin Ring ◽  
Residue Class Field ◽  
Image Position ◽  
Free Modules

The Betti numbers βn(k) of the residue class field k = R/m of a commutative local ring (R, m) have been studied for about 20 years, primarily as the coefficients of the Poincaré series of E . Several authors have obtained results about the growth of the sequence {βn(k)}.For example, Gulliksen [3] showed that when R is non-regular, the sequence is non-decreasing. More recently, Avramov [1] studied asymptotic properties of {βn(k)} and found that under certain conditions the growth is exponential, i.e., there is a natural number p such that for all n, βpn(k) ≧ 2n.In this paper, we examine the sequence {βn(M)} for arbitrary finitely generated non-free modules M over any commutative local artin ring R. We establish the following bounds:123where l(X) is the length of X.

The Maximal p-Extension of a Local Field

1971 ◽  
Vol 23 (3) ◽  
pp. 398-402 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Galois Group ◽  
Local Field ◽  
Residue Class ◽  
Prime Integer ◽  
Prime Field ◽  
Root Of Unity ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position ◽  
Relation Rank

1. Let k denote a local field, that is, a complete discrete-valued field with perfect residue class field . Let G denote the Galois group of the maximal separable algebraic extension M of k, and let g denote the corresponding object over . For a given prime integer p, let G(p) denote the Galois group of the maximal p-extension of k. The dimensions of the cohomology groupsconsidered as vector spaces over the prime field Z/pZ, are equal, respectively, to the rank and the relation rank of the pro-p-group G(p); see [4; 9]. These dimensions are well known in many cases, especially when k is finite [6; 3; (Hoechsmann) 2, pp. 297-304], but also when k has characteristic p, or when k contains a primitive pth root of unity [4, p. 205].

Lifting Isomorphisms of Modules

1979 ◽  
Vol 31 (4) ◽  
pp. 808-811 ◽  
Author(s):  
Irving Reiner
Keyword(s):  
Residue Class ◽  
Discrete Valuation Ring ◽  
Integral Group ◽  
Quotient Field ◽  
Finite Dimensional ◽  
Residue Class Field ◽  
Image Position ◽  
A Finite Group ◽  
Special Case ◽  
Reverse Implication

Throughout this note, let R be a discrete valuation ring with prime element π, residue class field , and quotient field K. Let Λ be an R-order in a finite dimensional K-algebra A. A Λ-lattice is an R-free finitely generated left Λ-module. For k > 0, we setwhere M is any Λ-lattice. Obviously, for Λ-lattices M and N,Maranda [1] and D. G. Higman [3] considered the reverse implication, and ProvedTHEOREM. Let Λ be an R-order in a separable K-algebra A. Then there exists a positive integer k (which depends on Λ) with the following property: for each pair of Λ-lattices M and N,Indeed,m it suffices to choose k so thatMaranda proved this result for the special case where Λ is the integral group ring RG of a finite group G.

scholarly journals On the Block of Defect Zero

1971 ◽  
Vol 44 ◽  
pp. 57-59 ◽  
Author(s):  
Yukio Tsushima
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Prime Divisor ◽  
Residue Class ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Residue Class Field ◽  
A Finite Group

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

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 Ramification Theory for Extensions of Degree p

1971 ◽  
Vol 41 ◽  
pp. 149-168 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Residue Class ◽  
Field Extension ◽  
Quotient Field ◽  
Class Field ◽  
Wild Ramification ◽  
Valuation Rings ◽  
Ramification Theory ◽  
Residue Class Field ◽  
Rank One

The notions of tame and wild ramification lead us to make the following definition.Definition. The quotient field extension of an extension of discrete rank one valuation rings is said to be fiercely ramified if the residue class field extension has a nontrivial inseparable part.

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

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.

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