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Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$,…

We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…

This paper contributes to the theory of orders of number fields. This paper defines a notion of "ray class group" associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including…

In this paper, we consider arithmetic progressions contained in Lucas sequences of first and second kind. We prove that for almost all sequences, there are only finitely many and their number can be effectively bounded. We also show that…

A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one…

For every prime number p, we show the existence of a solvable number field L ramified only at {p and infinity whose p-Hilbert Class field tower is infinite.

Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are…

Orders and fractional ideals in number fields provide interesting examples of lattices. We ask: what lattices arise from orders in number fields? We prove that all nontrivial multiplicative constraints on successive minima of orders come…

We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent M\"obius transformations. In particular, we obtain a full classification when $q$ is…

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting…

In this article we classify the complex quadratic number fields k with 2-class group of type (2,2,2) whose Hilbert 2-class fields have a 2-class group of rank 2, and then determine the length of their 2-class field towers.

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 the prime divisors in every class on a projective curve over a finite field contain arbitrary large truncated generalized classes of finite effective divisors.

Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…

We study the congruence classes attained by positive integers $D$ with a prescribed period of the continued fraction of $\sqrt D$. As an application, we refine the available results on large ranks of universal quadratic forms over real…

The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.

This paper is concerned with the problem of determining the number of division algebras which share the same collection of finite splitting fields. As a corollary we are able to determine when two central division algebras may be…

Some PARI programs have bringed out a property for the non-genus part of the class number of the imaginary quadratic fields, with respect to $(\sqrt D\,)^{\varepsilon}$, where $D$ is the absolute value of the discriminant and $\varepsilon…