Related papers: Classification of Algebraic Function Fields with C…

In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.

We show that two number fields with the same zeta function, and even with isomorphic adele rings, do not necessarily have the same class number.

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…

In the present paper we show that there are infinitely many classes of term functions in the free-void generated diagonalizable algebra, which are precomplete with respect to parametrical expressibility of functions.

A longstanding and important problem in algebraic geometry is the characterization of algebraic function fields. In this paper, we focus on the characterization problem for cyclotomic function field $L(\Lambda_M)$, which is an important…

We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.

We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P.Etingof it is proved that the number of semisimple Hopf algebras with a given…

We describe the isomorphism classes of certain infinite-dimensional graded Lie algebras of maximal class, generated by an element of weight one and an element of weight two, over fields of odd characteristic.

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.

In this paper the number of $\bar{\mathbb{F}}_q$-isomorphism classes of general Jacobi quartic curves, i.e., the number of general Jacobi quartic curves with distinct $j$-invariants, over the finite field $\mathbb{F}_q$ is enumerated.

Let $D$ be a division algebra over a field $F$. In this paper, we prove that there exist $a,b,x,y\in D^*$ such that $F(ab-ba)$ and $F(xyx^{-1}y^{-1})$ are maximal subfields of $D$, which answers questions posted in [5].

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2,2,2) whose Hilbert 2-class fields are finite.

We show that any multiplicative bijection between the algebras of differentiable functions, defined on differentiable manifolds of positive dimension, is an algebra isomorphism, given by composition with a unique diffeomorphism.

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

We study the automorphisms of a function field of genus $g\geq 2$ over an algebraically closed field of characteristic $p>0$. More precisely, we show that the order of a nilpotent subgroup $G$ of its automorphism group is bounded by $16…

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…

We extend to characteristic two recent results about isotropy of quadratic forms over function fields. In particular, we provide a characterization of function fields not only of quadratic forms but also more generally of polynomials in…

We complete the solution of the relative class number one problem for function fields of curves over finite fields. Using work from two earlier papers, this reduces to finding all function fields of genus 6 or 7 over $\mathbb{F}_2$ with one…

Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution…