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I. D. Chipchakov

Publications and source records attributed to I. D. Chipchakov.

14 recordsLinked to original sources
arXiv2018

Primarily quasilocal fields and 1-dimensional abstract local class field theory

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 group Br$(L/E)$ equals the maximal subgroup of Br$(E)$ of exponent $p$, for every cyclic extension $L/E$ of degree $p$. The paper proves that finite abelian extensions of $E$ are uniquely determined by their norm groups and related essentially as in the classical local class field theory. This includes analogues to the fundamental correspondence, the local reciprocity law and the local Hasse symbol.

math.RA↗
arXiv2015

On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions

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 absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in \mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) = \infty $, $p \dagger q ^{2} - q$, for every finitely-generated transcendental extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b _{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} = b _{2}$ and $0 \le b _{p} \le a _{p} \le \infty $, equals the sequence abrd$_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, for a field $E$ of characteristic zero.

math.RA↗
arXiv2015

On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions

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 $p$-dimension abrd$_{p}(E)$ is infinite, for some prime number $p$, it proves that for each pair $(n, m)$ of integers with $n \ge m > 0$, there is a central division $F$-algebra of Schur index $p ^{n}$ and exponent $p ^{m}$. Lower bounds on the Brauer $p$-dimension Brd$_{p}(F)$ are obtained in some important special cases where abrd$_{p}(E) < \infty $. These results solve negatively a problem posed by Auel et al. (Transf. Groups {\bf 16}: 219-264, 2011).

math.RA↗
arXiv2015

Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions

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 the field $F$ has Brauer $p$-dimension Brd$_{p}(F) \ge t$ except, possibly, in case $p = 2$, the Sylow pro-2-subgroups of $\mathcal{G}_{E}$ are of order 2, and $F$ is a nonreal field. It announces that Brd$_{p}(F)$ is infinite whenever $t \ge 1$ and the absolute Brauer $p$-dimension abrd$_{p}(E)$ is infinite; moreover, for each pair $(m, n)$ of integers with $1 \le m \le n$, there exists a central division $F$-algebra of exponent $p ^{m}$ and Schur index $p ^{n}$.

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arXiv2014

Henselian valued quasilocal fields with totally indivisible value groups, II

This paper characterizes the quasilocal fields from the class of Henselian valued fields with totally indivisible value groups, which possess finite separable extensions of nontrivial defect. We show that, for any prime number $q$, a divisible subgroup $T$ in the multiplicative group of complex numbers is realizable as the Brauer group of such a quasilocal field of residual characteristic $q$ unless $q = 2$ and the $2$-component of T$ is trivial.

math.RA↗
arXiv2011

Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions

This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$ does not possess nontrivial proper decomposition groups. When $r(p)_{E} = 2$, it describes the decomposition groups of $\mathcal{G}(E(p)/E)$. The paper shows that if $(K, v)$ is a $p$-Henselian valued field with $r(p)_{K} \in \mathbb N$ and a residue field of characteristic $p$, then $P \cong \widetilde P$ or $P$ is presentable as a semidirect product $\mathbb Z_{p}^τ \rtimes \widetilde P$, for some $τ\in \mathbb N$, where $\widetilde P$ is a Demushkin group of rank $\ge 3$ or a free pro-$p$-group. It also proves that when $\widetilde P$ is of the former type, it is continuously isomorphic to $\mathcal{G}(K ^{\prime}(p)/K ^{\prime})$, for some local field $K ^{\prime}$ containing a primitive $p$-th root of unity.

math.RA↗
arXiv2010

Algebraic extensions of global fields admitting one-dimensional local class field theory

Let $E$ be an algebraic extension of a global field $E_{0}$ with a nontrivial Brauer group Br$(E)$, and let $P(E)$ be the set of those prime numbers $p$, for which $E$ does not equal its maximal $p$-extension $E(p)$. This paper shows that $E$ admits one-dimensional local class field theory if and only if there exists a system $V(E) = \{v(p)\colon \ p \in P(E)\}$ of (nontrivial) absolute values, such that $E(p) \otimes_{E} E_{v(p)}$ is a field, where $E_{v(p)}$ is the completion of $E$ with respect to $v(p)$. When this occurs, we determine by $V(E)$ the norm groups of finite extensions of $E$, and the structure of Br$(E)$. It is also proved that if $P$ is a nonempty set of prime numbers and $\{w(p)\colon \ p \in P\}$ is a system of absolute values of $E_{0}$, then one can find a field $K$ algebraic over $E_{0}$ with such a theory, so that $P(K) = P$ and the element $κ(p) \in V(K)$ extends $w(p)$, for each $p \in P$.

math.NT↗
arXiv2009

On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions

This paper shows that divisible abelian torsion groups are realizable as Brauer groups of quasilocal fields. It describes the isomorphism classes of Brauer groups of primarily quasilocal fields and solves the analogous problem concerning the reduced components of the Brauer groups of two basic types of Henselian valued absolutely stable fields. For a quasilocal field E and a finite separable extension R/E, we find two sufficient conditions for validity of the norm group equality $N(R/E) = N(R_{0}/E)$, where R_{0} is the maximal abelian extension of E in R. This is used for deriving information on the arising specific relations between Galois groups and norm groups of finite Galois extensions of E.

math.RA↗
arXiv2008

Norm groups and class fields of formally real quasilocal fields

This paper establishes a relationship between finite extensions and norm groups of formally real quasilocal fields, which yields a generally nonabelian local class field theory, including analogues to the fundamental correspondence, the local reciprocity law and the norm limitation theorem.

math.RA↗
arXiv2007

On the residue fields of Henselian valued stable fields

Let $(K, v)$ be a Henselian valued field satisfying the following conditions, for a given prime number $p$: (i) central division $K$-algebras of (finite) $p$-primary dimensions have Schur indices equal to their exponents; (ii) the value group $v(K)$ properly includes its subgroup $pv(K)$. The paper shows that if $\hat K$ is the residue field of $(K, v)$ and $\hat R$ is an intermediate field of the maximal $p$-extension $\hat K (p)/\hat K$, then the natural homomorphism Br$(\hat K) \to $ Br$(\hat R)$ of Brauer groups maps surjectively the $p$-component Br$(\hat K)_{p}$ on Br$(\hat R)_{p}$. It proves that Br$(\hat K)_{p}$ is divisible, if $p > 2$ or $\hat K$ is a nonreal field, and that Br$(\hat K)_{2}$ is of order 2 when $\hat K$ is formally real. We also obtain that $\hat R$ embeds as a $\hat K$-subalgebra in a central division $\hat K$-algebra $\hat Δ$ if and only if the degree $[\hat R\colon \hat K]$ divides the index of $\hat Δ$.

math.RA↗
arXiv2007

On the residue fields of Henselian valued stable fields, II

Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and $D$ a central division $E$-algebra of index divisible by $[M\colon E]$. In addition to the main result of Part I, this part of the paper shows that if the Galois group $G(M/E)$ is not nilpotent, then $M$ does not necessarily embed in $D$ as an $E$-subalgebra. When $E$ is quasilocal, we find the structure of the character group of its absolute Galois group; this enables us to prove that if $E$ is strictly quasilocal and almost perfect, then the divisible part of the multiplicative group $E ^{\ast}$ equals the intersection of the norm groups of finite Galois extensions of $E$.

math.RA↗
arXiv2005

On the scope of validity of the norm limitation theorem for quasilocal fields

Let $E$ be a quasilocal field, $R/E$ a finite separable extension, 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, if the natural Brauer group homomorphism Br$(E) \to $ Br$(L)$ is surjective, for every finite extension $L$ of $E$. The paper proves in a strong form that the surjectivity of the homomorphisms Br$(E) \to $ Br$(L)$ is essential.

math.RA↗
arXiv2005

On the scope of validity of the norm limitation theorem in one-dimensional abstract local class field theory

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 following two special cases: (i) $E$ is primarily quasilocal and $R$ is an intermediate field of a finite Galois extension $M/E$ with a nilpotent Galois group; (ii) $E$ is quasilocal and the natural Brauer group homomorphism Br$(E) \to $ Br$(L)$ is surjective, for every finite extension $L$ of $E$. It is also used for describing the norm groups of formally real quasilocal fields, and of Henselian discrete valued fields whose finite extensions are strictly PQL. The paper proves that the condition on $G(M/E)$ in (i) cannot be weakened, and the surjectivity of the homomorphism Br$(E) \to $ Br$(L)$ is essential for the validity of (ii).

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