Searcharxiv⌕ Search

arXiv subjects

Sergey V. Tikhonov

Publications and source records attributed to Sergey V. Tikhonov.

7 recordsLinked to original sources
arXiv2025

Polynomials over division rings

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root of a polynomial over an arbitrary division ring, then the conjugacy class of this element contains infinitely many elements that are not roots of this polynomial. The paper also contains estimates for the number of different conjugacy classes of spherical roots for some types of polynomials over quaternion division algebras.

math.RA↗
arXiv2024

Genus of division algebras over fields with infinite transcendence degree

We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let $K$ be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if $D$ is a central division $K$-algebra, then ${\bf gen}(D)$ consists of Brauer classes $[D']$ such that $[D]$ and $[D']$ generate the same subgroup of $Br(K)$. In particular, the genus of any division $K$-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if $char(K) \ne 2$, we prove that the genus of a simple algebraic group of type $\mathrm{G}_2$ over such a field $K$ is trivial.

math.RA↗
arXiv2023

Outer forms of type $A_2$ with infinite genus

Let $G$ be an absolutely almost simple algebraic group over a field $K$. The genus ${\bf gen}_K(G)$ of $G$ is the set of $K$-isomorphism classes of $K$-forms $G'$ of $G$ that have the same $K$-isomorphism classes of maximal $K$-tori as $G$. We construct an example of outer forms of type $A_2$ with infinite genus.

math.RA↗
arXiv2020

On genus of division algebras

The genus $gen(D)$ of a finite-dimensional central division algebra $D$ over a field $F$ is defined as the collection of classes $[D']\in Br(F)$, where $D'$ is a central division $F$-algebra having the same maximal subfields as $D$. We show that the fact that quaternion division algebras $D$ and $D'$ have the same maximal subfields does not imply that the matrix algebras $M_l(D)$ and $M_l(D')$ have the same maximal subfields for $l>1$. Moreover, for any odd $n>1$, we construct a field $L$ such that there are two quaternion division $L$-algebras $D$ and $D'$ and a central division $L$-algebra $C$ of degree and exponent $n$ such that $gen(D) = gen(D')$ but $gen(D \otimes C) \ne gen(D' \otimes C)$.

math.RA↗
arXiv2014

Division algebras of prime degree with infinite genus

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D'] in the Brauer group Br(F), where D' is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p, and any two such algebras have the same genus.

math.RA↗