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Ivan D. Chipchakov

Publications and source records attributed to Ivan D. Chipchakov.

10 recordsLinked to original sources
arXiv2025

Locally finite-dimensional central division algebras over function fields of curves over m-local fields, criterion for normality

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 $1$. The paper under review shows that if $K _{0}$ is a field of finite Diophantine dimension ddim$(K _{0})$, and $R$ is an associative locally finite-dimensional central division $K$-algebra, then $R$ is a normally locally finite algebra over $K$, that is, every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R}_{Y}$ of $R$.

math.NT↗
arXiv2024

On K\"{o}the's normality question for locally finite-dimensional central division algebras

This paper considers K\"{o}the's question of whether every associative locally finite-dimensional (abbr., LFD) central division algebra $R$ over a field $K$ is a normally locally finite (abbr., NLF) algebra over $K$, that is, whether every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R} _{Y}$ of $R$. It shows that the answer to the posed question is negative if $K$ is a purely transcendental extension of infinite transcendence degree over an algebraically closed field $k$. On the other hand, central division LFD-algebras over $K$ turn out to be NLF in the following special cases: (i) $K$ is a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $K _{0}$; (ii) $K$ is a higher-dimensional local field with last residue field equal to $K _{0}$.

math.RA↗
arXiv2023

Quasifinite fields of prescribed characteristic and Diophantine dimension

Let $\mathbb{P}$ be the set of prime numbers, $\overline {\mathbb{P}}$ the union $\mathbb{P} \cup \{0\}$, and for any field $E$, let char$(E)$ be its characteristic, ddim$(E)$ the Diophantine dimension of $E$, $\mathcal{G}_{E}$ the absolute Galois group of $E$, and cd$(\mathcal{G}_{E})$ the Galois cohomological dimension $\mathcal{G}_{E}$. The research presented in this paper is motivated by the open problem of whether cd$(\mathcal{G}_{E}) \le {\rm ddim}(E)$. It proves the existence of quasifinite fields $Φ_{q}\colon q \in \mathbb{P}$, with ddim$(Φ_{q})$ infinity and char$(Φ_{q}) = q$, for each $q$. It shows that for any integer $m > 0$ and $q \in \overline {\mathbb{P}}$, there is a quasifinite field $Φ_{m,q}$ such that char$(Φ_{m,q}) = q$ and ddim$(Φ_{m,q}) = m$. This is used for proving that for any $q \in \overline {\mathbb{P}}$ and each pair $k$, $\ell \in (\mathbb{N} \cup \{0, \infty \})$ satisfying $k \le \ell $, there exists a field $E _{k, \ell ; q}$ with char$(E _{k, \ell ; q}) = q$, ddim$(E _{k, \ell ; q}) = \ell $ and cd$(\mathcal{G}_{E_{k, \ell ; q}}) = k$. Finally, we show that the field $E _{k, \ell ; q}$ can be chosen to be perfect unless $k = 0 \neq \ell $.

math.NT↗
arXiv2022

On algebraic central division algebras over Henselian fields of finite absolute Brauer $p$-dimensions and residually arithmetic type

Let $(K, v)$ be a Henselian field with a residue field $\widehat K$ and value group $v(K)$, and let $\mathbb{P}$ be the set of prime numbers. This paper finds conditions on $K$, $v(K)$ and $\widehat K$ under which every algebraic associative central division $K$-algebra $R$ contains a central $K$-subalgebra $\widetilde R$ decomposable into a tensor product of central $K$-subalgebras $R _{p}$, $p \in \mathbb{P}$, of finite $p$-primary dimensions $[R _{p}\colon K]$, such that each finite-dimensional $K$-subalgebra $\Delta $ of $R$ is isomorphic to a $K$-subalgebra $\widetilde \Delta $ of $\widetilde R$.

math.RA↗
arXiv2022

Henselian discrete valued stable fields

Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $q \ge 0$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$, for each prime $p$. The present paper shows that if $p = q$, then Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is a $p$-quasilocal field and the degree $[\widehat K\colon \widehat K ^{p}]$ is $\le p$. This complements our earlier result that, in case $p \neq q$, we have Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is $p$-quasilocal and Brd$_{p}(\widehat K) \le 1$.

math.RA↗
arXiv2021

Fields of dimension one algebraic over a global or local field need not be of type $C_{1}$

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$(E ^{\prime })$ is trivial, for every algebraic extension $E ^{\prime }/E$; (ii) finite extensions of $E$ are not $C _{1}$-fields. This, applied to the maximal algebraic extension $K$ of the field $\mathbb{Q}$ of rational numbers in the field $\mathbb{Q} _{p}$ of $p$-adic numbers, for a given prime $p$, proves the existence of an algebraic extension $E _{p}/\mathbb{Q}$, such that dim$(E _{p}) \le 1$, $E _{p}$ is not a $C _{1}$-field, and $E _{p}$ has a Henselian valuation of residual characteristic $p$.

math.NT↗
arXiv2019

On index-exponent relations over Henselian fields with local residue fields

Let $p$ be a prime number and $(K, v)$ a Henselian valued field with a residue field $\widehat K$. This paper determines the Brauer $p$-dimension of $K$, in case $p \neq {\rm char}(\widehat K)$ and $\widehat K$ is a $p$-quasilocal field properly included in its maximal $p$-extension. When $\widehat K$ is a local field, it describes index-exponent pairs of central division $K$-algebras of $p$-primary degrees. The same goal is achieved, if $(K, v)$ is maximally complete, char$(K) = p$ and $\widehat K$ is local.

math.RA↗
arXiv2018

On Brauer $p$-dimensions and absolute Brauer $p$-dimensions of Henselian fields

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 $\widehat K$, such as the condition abrd$_{p}(\widehat K) = 0$. It describes the set $Σ_{0}$ of sequences ${\rm abrd}_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, where $\mathbb P$ is the set of prime numbers and $E$ runs across the class of Henselian fields with char$(\widehat E) = 0$ and a projective absolute Galois group $\mathcal{G}_{\widehat E}$. Specifically, $Σ_{0}$ contains a sequence $a_{p}, b_{p} \in \mathbb N \cup \{0, \infty \}$, $p \in \mathbb P$, whenever $a_{2} \le 2b_{2}$ and $a_{p} \ge b_{p}$, for each $p$. Similar results are obtained in characteristic $q > 0$.

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