Related papers: Algebraically closed $\sigma$-fields

In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its…

Let K be an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. Let X be a K-rigid analytic variety and \Sigma a semianalytic subset of X. Then the closure of \Sigma in X with respect to the…

We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of…

We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism…

In constructive algebra one cannot in general decide the irreducibility of a polynomial over a field K. This poses some problems to showing the existence of the algebraic closure of K. We give a possible constructive interpretation of the…

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all…

Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the…

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a…

Let $S$ be a unital associative ring and $S[t;\sigma,\delta]$ be a skew polynomial ring, where $\sigma$ is an injective endomorphism of $S$ and $\delta$ a left $\sigma$-derivation. For each $f\in S[t;\sigma,\delta]$ of degree $m>1$ with a…

In this paper we will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in…

Classification, up to isomorphism, of algebras from a non-empty subset of the variety of $n$- dimensional algebras is presented. It is shown that these algebras have only trivial automorphism and if the basic field is algebraically closed…

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…

We give a criterion when a polynomial $x^n-g$ is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic.

In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for…

Building over recent results, we expand the basic theory of algebraic extensions to the realm of superfields -a field with multivalued sum and product-, showing that every superfield has a (unique up to isomorphism) strong algebraic…

We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several…

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full…

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1…

Let $A$ be a ring and $\sigma: A \to A$ a ring endomorphism. A generalized skew (or $\sigma$-)derivation of $A$ is an additive map $d: A \to A$ for which there exists a map $\delta:A \to A$ such that $d(xy)=\delta(x)y+\sigma(x)d(y)$ for all…

Let $R$ be a ring and $\sigma$ an endomorphism of $R$. In this note, we study skew polynomial rings and skew power series rings over idempotent reflexive rings and abelian rings. Also, we introduce the concept of right (resp., left)…