Related papers: Algebraic extensions of global fields admitting on…

Let $E$ be a field satisfying the following conditions: (i) the $p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$ is a prime number for which $E$ is properly included in its maximal $p$-extension; (ii) the relative Brauer…

Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ satisfying the following: (i) $E$ has dimension dim$(E) \le 1$, i.e. the Brauer group Br$(E…

Let $K$ be a global or local field, $E/K$ a Galois extension, and Br$(E)$ the Brauer group of $E$. This paper shows that if $K$ is a local field, $v$ is its natural discrete valuation, $v'$ is the valuation of $E$ extending $v$, and $q$ is…

Let $E$ be a field, $p$ a prime number and $F/E$ a finitely-generated extension of transcendency degree $t$. This paper shows that if the absolute Galois group $\mathcal{G}_{E}$ is of nonzero cohomological $p$-dimension cd$_{p}(E)$, then…

Given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n=2. As a…

The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite…

Let $E$ be a field, $R$ a finite separable extension of $E$, and $R_{\rm ab}$ the maximal abelian subextension of $E$ in $R$. The main result of this paper shows that the norm groups $N(R/E)$ and $N(R_{\rm ab}/E)$ are equal in each of the…

Let $E$ be a field of absolute Brauer dimension abrd$(E)$, and $F/E$ a transcendental finitely-generated extension. This paper shows that the Brauer dimension Brd$(F)$ is infinite, if abrd$(E) = \infty $. When the absolute Brauer…

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions,…

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois…

Let $P$ be a Poisson algebra, $E$ a vector space and $\pi : E \to P$ an epimorphism of vector spaces with $V = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Poisson algebra structures that can be defined…

Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility…

Let $K$ be a field complete with respect to a nonarchimedean real-valued norm, and let $L/K$ be an algebraic extension. We show that there is a unique norm on $L$ extending the given norm on $K$, with an explicit description. As an…

There is a known analogy between growth questions for class groups and for Selmer groups. If $p$ is a prime, then the $p$-torsion of the ideal class group grows unboundedly in $\mathbb{Z}/p\mathbb{Z}$-extensions of a fixed number field $K$,…

Let $K _{m}$ be an $m$-local field with an $m$-th residue field $K _{0}$, for some integer $m > 0$, and let $K/K _{m}$ be a field extension of transcendence degree trd$(K/K _{m}) \le 1$. This paper shows that if $K _{0}$ is a field of…

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \land p = q$. For a logic $L$ algebraized by a quasivariety $\mathcal{Q}$ we show that the AE-subclasses of…

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

This paper determines the Brauer $p$-dimension Brd$_{p}(K)$ and the absolute Brauer $p$-dimension abrd$_{p}(K)$ of a Henselian valued field $(K, v)$, for a prime $p \neq {\rm char}(\widehat K)$, under restrictions on the residue field…

Author's generalization of one-dimensional class field theory to theory of abelian totally ramified p-extensions of a complete discrete valuation field with arbitrary non-separably p-closed residue field and its applications are described.