Soft Int-Field Extension

The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties.

Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials

Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a 0 , . . ., an− 1, where n is the degree of the minimal polynomial of a over F .

Strongly and co-strongly minimal abelian structures

We give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

Norm and spectral radius for algebraic elements of a Banach algebra

The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formulacontains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,(see Theorem 2), while for a general Banach algebra we have at least

Defining algebraic elements

This paper is a survey and a synthesis of the various approaches to defining algebraic elements. Most of it is devoted to proving the following result.Let T be a universal theory with the Amalgamation Property. Then for T the notions of algebraic element introduced by Robinson, Jónsson, and Morley are identical. Furthermore, they extend in a natural way the notion of algebraic element introduced by Park, and used by Lachlan and Baldwin and by Kueker.In the course of proving this we shall construct the algebraic closure as a suitable injective hull and prove a unique factorisation theorem for algebraic predicates.We shall also show (in §3) that if T is closed under products then algebraic elements all have degree 1. Thus in algebra, algebraic elements reduce to epimorphisms.To demonstrate the remarkable stability of the notion we shall show (at the end of §5) that defining algebraic elements by infinitary formulas yields no new ones.Let L be a language. The cardinality of the set of formulas of L is denoted by ∣L∣. An L-theory is a deductively closed set of L-sentences. We let denote the category of models of T and (L-structure) homomorphisms between them. If A is a substructure of B we write A ≤ B. We call u: A → B an injection if u is an isomorphism of A with a substructure of B, and let denote the subcategory of consisting of all injections.