On the Brauer p-dimension of Henselian discrete valued fields of residual characteristic p > 0

Let be a Henselian discrete valued field with residue field of characteristic , and be the Brauer p-dimension of K. This paper shows that if , for some . It proves that if and only if .     

Givens' Transformation Applied to Quaternion Valued Vectors

Givens' transformation (1954) was originally applied to real matrices. We shall give an extension to quaternion valued matrices. The complex case will be treated in the introduction. We observe that the classical Givens' rotation in the real and in the complex case is itself a quaternion using an isomorphism between certain (2×2) matrices and R ⁴ equipped with the quaternion multiplication. In the real and complex case Givens' (2×2) matrix is determined uniquely up to an arbitrary (real or complex) factor with ||=1. However, because of the noncommutativity of quaternions, we shall show that in the quaternion case such a factor must obey certain additional restrictions. There are two numerical examples including a MATLAB program and some hints for implementation. Matrices with quaternion entries arise, e.g., in quantum mechanics.This revised version was published online in October 2005 with corrections to the Cover Date.     

Note on a Variation of the Schröder-Bernstein Problem for Fields

In this note we study fields F with the property that the simple transcendental extension F(u) of F is isomorphic to some subfield of F but not isomorphic to F. Such a field provides one type of solution of the Schröder-Bernstein problem for fields.     

具有卷积的中心自反环

本文研究了具有卷积的中心自反环的性质,定义并引入了中心-自反环,显然,中心-自反环是自反环、中心自反环和-自反环的推广.给出了这类环的一些特征,研究了相关的环扩张,包括平凡扩张,Dorroh扩张和多项式扩张.     

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.     

Stable valued fields

We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Güntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using — as in the general theory of valuations — the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields. Supported by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools via project NSh-4787.2006.1 and by RFBR grant No. 05-01-00481. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 707–728, November–December, 2007.     

广义四元数体上矩阵的最小多项式

本文给出了广义四元数体上方阵的最小多项式与最小中心多项式的构造公式,讨论了它们的性质及其应用,得到广义四元数方阵相似于对角矩阵的一个充要条件。     

Single-valued extension property at the points of the approximate point spectrum

A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point in the case that λ_oI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

Operators Which Do Not Have the Single Valued Extension Property

In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T  L(X) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ_o  C.     

An Involutorial Version of a Theorem of Gräter

Let D be a division ring with centre Z and with involution (*). Let V be a valuation of D with value group Γ, a linearly ordered additive group (non necessarily commutative) together with a symbol ∞ (positive infinity). We assume that for each nonzero symmetric element s = s* of D, which is algebraic over Z, we have for all nonzero elements x of D, V(xa ? ax) > V(ax). We define the residue characteristic exponent p of V to be the characteristic χ of the associated residue division ring written as D _V , if χ ≠ 0, and p = 1, if χ = 0. We show here that if F is a finite dimensional commutative subalgebra of D over Z, which is *-closed (i.e., F* = F), and if (*) is of the first kind (i.e., each central element of D must be symmetric), then [F: Z] = 2 ^r p ^m where m is a nonnegative integer and r = 0 or 1 according as the restricted involution in F is trivial or not. The case of an involution (*) of the second kind (i.e., some central element of D is not symmetric) requires (for this author) a stronger type of valuation, namely, V is a *-valuation, that is to say, for all elements x of D, we have V(x*) = V(x), a condition which readily implies Γ must be Abelian. Here, we can show that for F as in the preceding, [F: Z] = p ^m , where m is again a nonnegative integer. The preceding results generalize a theorem of Gräter and improve in parts recent theorems of this author in [2 Chacron , M. ( 2014 ). Residually *-abelian valuation with residue characteristic not 2 . Communication in Algebra 42 : 2956 – 2968 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In the special case p = 2 the results provide a modicum of answers to the questions opened informally in [2 Chacron , M. ( 2014 ). Residually *-abelian valuation with residue characteristic not 2 . Communication in Algebra 42 : 2956 – 2968 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] (see concluding paragraph in [2 Chacron , M. ( 2014 ). Residually *-abelian valuation with residue characteristic not 2 . Communication in Algebra 42 : 2956 – 2968 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] or here Question 3.2.1). More is to be said in the third and final section of this work.     

A boolean transfer principle from L*-Algebras to AL*-Algebras

Just as Kaplansky [4] has introduced the notion of an AW*-module as a generalization of a complex Hilbert space, we introduce the notion of an AL*-algebra, which is a generalization of that of an L*-algebra invented by Schue [9, 10]. By using Boolean valued methods developed by Ozawa [6–8], Takeuti [11–13] and others, we establish its basic properties including a fundamental structure theorem. This paper should be regarded as a continuation or our previous paper [5], the familiarity with which is presupposed. MSC: 03C90, 03E40, 17B65, 46L10.     

Stability preservation theorems

The basic result of the paper is the main theorem worded as follows. Let {ie155-01} be a valued field such that {ie155-02} has characteristic p > 0 and let {ie155-03} be an extension of valued fields satisfying the following conditions: (i) there exists a set {ie155-04} for which {ie155-05} is a separating transcendence basis for a field {ie155-06} over F_R; (ii) Γ _R is p-pure in {ie155-07}, i.e., {ie155-08} does not contain elements of order p; (iii) there exists a set B₁ ⊂ F₀^× such that the family {ie155-09} is linearly independent in the elementary p-group {ie155-10}; (iv) F₀ is algebraic over F(B₀ ⋃ B₁). Then the property of being stable for {ie155-11} implies being stable for {ie155-12}. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and by RFBR (grant No. 08-01-00442-a). __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 269–287, May–June, 2008.     

Deformations of the central extension of the Poisson superalgebra

The Poisson superalgebra realized on smooth Grassmann-valued functions with compact support has a central extension at some values of the superdimension. We find formal deformations of these central extensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 250–269, August, 2008.