Yu. L. Ershov

Model of Partial Continuous Functionals

In the last years (most fully in the thesis by M. Hyland [6]) the coincidence of a large number of models of everywhere defined functionals of the finite type has been proved. This coincidence demonstrates the definite significance of the model , which is the model of the bar-recursive functionals theory as well. However, I am convinced that the most natural way of this model determination is in determining it by the natural model of partial continuous functionals, and that it is model that is more fundamental than model , in the general mathematical sense. The most essential point is that partial objects may be built by the successive expansion of finite objects, which are the elements of model . The history of recursive functions theory, in particular, shows that only after the introduction of the notion “partially recursive function” into usage this theory acquired well-composed structure. Further account is divided into two parts. The first and larger part contains the definitions and the properties of some classes of topological spaces, the efficience of which, I presume, is not covered by the construction of model or the models of λ-calculus [2]. The second part contains the main results concerning model .

Chapter 5 Σ-Definability of algebraic structures

Publisher Summary This chapter discusses the Σ-definability of an algebraic structure. The notion of the Σ-definability of an algebraic structure in an admissible set was introduced as a generalization of the notion of constructivization. Σ-definability of an algebraic structure is illustrated through notions and facts from the theory of admissible sets or KPU-models. It is useful to obtain information beforehand on definability in such structures. The chapter also discusses an example of a complete decidable theory only the simple model of which is locally constructivizable.

Fields with continuous local elementary properties. III

This article is a direct continuation of [1], a familiarity with the results of which is desirable. We will be concerned with a proof of an analog of Theorem 3 from [1] to extend it to the case of fields in RCE* (with absolutely unramified valuation rings) possessing a “large” absolute Galois Δ-group.

Algebraically compact groups

First, some preliminaries. Let us recall the definition of the p-height of an element in a group. If p is a prime and ~£A , then by the ~-height of ~ in the group ~ we mean the ordinal ~7~a)~44~{~ I ~ ~), and the equation p~= a is solvable in A}. We have ~ [a)~---~ U) f --~;(0) =U). We state s number of properties of the ~-height which always are well known and can be verified directly. The upper index will often be omitted when it is clear which group we have in mind.

The number of linear orders on a field

An example is given of a field which can be ordered in exactly ℵ ways where ℵ is a given cardinal number.

Tame and purely wild extensions of valued fields

A systematic and concise exposition of the basic results concerning two complementary classes (tame and purely wild) of extensions of (Henselian) valued fields is given. These notions proved to be quite useful both for the general theory and for the model theory of such fields. Along with new results, new proofs of old results are presented. Thus, in the proof of the well-known Pank theorem on the existence of a complement to the ramification group in the absolute Galois group of a Henselian valued field, the properties of maximal immediate extensions are employed instead of cohomological methods. Let F be a field. By definition, a valuation ring of F is an arbitrary subring R of F such that, for each a ∈ F× = F \ {0}, we have either a ∈ R or a−1 ∈ R. Every valuation ring R is a local ring; i.e., it has a unique maximal ideal m(R). The quotient ring FR R/m(R) of R by the maximal ideal m(R) is called the residue field. The quotient group ΓR F×/U(R) of the multiplicative group F× of the field F by the multiplicative group U(R) of units (invertible elements) of R is called the valuation group. The group ΓR is equipped with the structure of an ordered group in which the cone of nonnegative elements is the image of the set R \ {0} in ΓR. We use the additive notation for the operation in ΓR. The homomorphism vR : F× → ΓR = F×/U(R) is called the valuation (determined by the valuation ring R). A pair F = 〈F, R〉, where F is a field and R is a valuation ring of F , is called a valued field. If F = 〈F, R〉, F0 = 〈F0, R0〉 are valued fields, F ≤ F0, and R0 ∩ F = R, then F0 is called an extension of F (F ≤ F0). If F ≤ F0, then the residue field FR is naturally identified with a subfield of the residue field FR0 (FR ≤ FR0), and the valuation group ΓR is naturally identified with a subgroup of ΓR0 (ΓR ≤ ΓR0). A valued field F is said to be Henselian if, for every algebraic extension F0 ≥ F , there exists a unique valuation ring R0 in F0 such that 〈F0, R0〉 ≥ F. A valued field F = 〈F, R〉 is Henselian if and only if the following statement is valid. Hensel lemma. Let f ∈ R[x] be a monic polynomial such that the image f̄ of f in FR[x] has a simple root ᾱ in FR. Then the polynomial f has a root α ∈ R such that ᾱ = α + m(R). If F0 ≥ F is a finite extension (i.e., [F0 : F ] < ω) of valued fields, then the degree f [FR0 : FR] of the extension FR0 ≥ FR and the index e [ΓR0 : ΓR] of the subgroup ΓR in ΓR0 are finite and we have the following basic inequality: e · f ≤ n [F0 : F ]. 2000 Mathematics Subject Classification. Primary 12F15.

Fields with two linear orderings

We characterize fields which are maximal with respect to the property of having two different linear orderings. The Galois group of the algebraic closure of a maximal field is described. An example of non-uniqueness of the maximal extension is mentioned.

ON AN ARTICLE BY R. BROWN

We establish a sufficient condition for the existence of a root of a polynomial over a Henselian valued field, extending the main result of the article [1] by Brown.

SEPARANTS OF SOME POLYNOMIALS

The explicit form is given of the separants of the polynomials of the author’s previous article [1]. This entails clarification of the main theorem of [1].

LOCAL CLASS FIELD THEORY

New sufficient conditions for the validity of local class field theory for Henselian valued fields are established. An example is presented to show that these conditions are less restrictive than the applicability of the Neukirch abstract class field theory.