Hensel赋值域

1°Hensel赋值域的理论起源于Kurt Hensel在他的p-进数论中所建立的一条引理,这条引理是对p-进数域上多项式的可分解性所作的一个判别法则。在赋值论出现以后,Rychlik把它推广到一阶赋值的完全域上,从而在赋值理论中引进了一个重要的代数方法。三十年代,Krull对任意阶的赋值引入了完全性的概念,这个重要的引理又在任意阶的完全域上建立起来。但是,这个引理在赋值理论中的重要性并不依赖于赋值域的完全性,首先认识到这个事实的是瑞士数学家Alexander Ostronwski.现在我们遵循他的方式来作如下的规定:     

Extremal Valued Fields

It is shown that every finite-dimensional skew field whose center is an extremal valued field is defect free. We construct an example of an algebraically complete valued field such that a finite-dimensional skew field over it has a non-trivial defect, that is, there exist algebraically complete valued fields that are not extremal.     

A New Criterion on k-Normal Elements over Finite Fields

The notion of normal elements for finite fields extension was generalized as k-normal elements by Huczynska et al.(2013). Several methods to construct k-normal elements were presented by Alizadah et al.(2016) and Huczynska et al.(2013), and the criteria on k-normal elements were given by Alizadah et al.(2016) and Antonio et al.(2018). In the paper by Huczynska, S., Mullen, G., Panario, D. and Thomson, D.(2013), the number of k-normal elements for a fixed finite field extension was calculated and...     

Self-pure-generators over Dedekind domains

We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind domain are A-generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that $\mathbf{type}(B)$ is the inner type of A. This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.     

Motivic cohomology over Dedekind rings

We study properties of Blochs higher Chow groups on smooth varieties over Dedekind rings. We prove the vanishing of for i > n, and the existence of a Gersten resolution for if the residue characteristic is p. We also show that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture an identification for m invertible, and a Gersten resolution with (arbitrary) finite coefficients. Over a complete discrete valuation ring of mixed characteristic (0,p), we construct a map from motivic cohomology to syntomic cohomology, which is a quasi-isomorphism provided the Bloch-Kato conjecture holds.Supported in part by JSPS, NSF Grant. No. 0070850, and the Alfred P.Sloan Foundation     

Valued Fields with K Commuting Derivations

A matrix is said to be monomial if every row and column has only one nonzero entry. Let G be a group. A representation ρ: G → GL _n (?) is said to be a monomial representation of G if there exists a basis with respect to which ρ(g) is a monomial matrix for every g ∈ G. We use elementary methods to classify the irreducible monomial representations of the groups L ₂(q), L ₃(q) and their natural decorations.     

Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

ABSTRACT

Let ? be a complete set of Sylow subgroups of a finite group G, that is, ? contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be ?-permutable if H permutes with every member of ?. The purpose of this article is to study the influence of ?-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998 Asaad , M. ( 1998 ). On maximal subgroups of Sylow subgroups of finite groups . Comm. Algebra 26 ( 11 ): 3647 – 3652 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Asaad and Heliel (2003 Asaad , M. , Heliel , A. A. ( 2003 ). On permutable subgroups of finite groups . Arch. Math. 80 : 113 – 118 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Asaad et al. (1991 Asaad , M. , Ramadan , M. , Shaalan , A. ( 1991 ). Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group . Arch. Math. 56 : 521 – 527 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Li et al. (2003 Li , Y. , Wang , Y. , Wei , H. ( 2003 ). The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group . Arch. Math. 81 ( 3 ): 245 – 252 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Ramadan (1992 Ramadan , M. ( 1992 ). Influence of normality on maximal subgroups of Sylow subgroups of a finite group . Acta Math. Hungar. 59 ( 1–2 ): 107 – 110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), and Srinivasan (1980 Srinivasan , S. ( 1980 ). Two sufficient conditions for supersolvability of finite groups . Israel J. Math. 35 : 210 – 214 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]).     

Group rings over Dedekind rings

Seghal posed the following question: IfA andB are rings, doesA[X,X ⁻¹] ℞B[X,X ⁻¹] implyA ℞B. In general the answer to this question is no. In this note we give an affirmative answer in the case thatA andB are Dedekind rings. The author is research assistant at the NFWO.     

Torsion-free modules over Dedekind rings

We obtain necessary and sufficient conditions in order that an arbitrary pure monoendomorphism of a module decomposed into a direct sum of rank 1 torsion-free modules over a Dedekind ring be an automorphism.     

Exchanging torsion modules over Dedekind domains

The author was partially supported by a grant (Project 1419) from the Research Grants Committee of the University of Alabama.     

On Secondary Modules over Dedekind Domains

The aim of this paper is to classify the indecomposable secondary modules over local dedekind domains and to establish a connection between the indecomposable secondary modules and the indecomposable pure-injective modules over such domains.     

Dedekind整环上的不可分解扭模

We give a direct proof to the classification of all indecomposable torsion modules over a Dedekind ring. As an application, we classify all indecomposable locally-nilpotent modules over the polynomial algebra with one variable over a field.     

Irreducibility Criterion Over Finite Fields

The aim of this article is to prove the irreducibility of the polynomial Λ(Y) = Y ^d  + λ _d?1 Y ^d?1 + … + λ₀ over 𝔽 _q [X] where λ _i ∈ 𝔽 _q [X] and deg λ _d?1 > deg λ _i for each i ≠ d ? 1. We discuss in particular connections between the irreducible polynomials Λ and the number of Pisot elements in the case of formal power series.