Related papers: Simplest cubic fields with small class number

For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a…

Let $M$ be a subset of $\mathbb{R}^k$. It is an important question in the theory of linear inequalities to estimate the minimal number $h=h(M)$ such that every system of linear inequalities which is infeasible over $M$ has a subsystem of at…

Let ${\mathbb{F}_{q}}$ be the finite field of order $q$. Let $G$ be one of the three groups ${\rm GL}(n, \mathbb{F}_q)$, ${\rm SL}(n, \mathbb{F}_q)$ or ${\rm U}(n, \mathbb{F}_q)$ and let $W$ be the standard $n$-dimensional representation of…

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m…

We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a given bound.

Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show that there are infinitely many number fields of degree $D$ whose primitive elements all have relatively large height in terms of $b$, $D$ and the discriminant of…

A representation field for a non-maximal order H in a central simple algebra A is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we…

We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any…

Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where…

We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic form over any $p$-adic field, provided the cardinality of the residue class field exceeds 293. That is any Cubic and Quadratic form with at least 14 variables has a…

We show that the minimal number of skewed hyperplanes that cover the hypercube $\{0,1\}^{n}$ is at least $\frac{n}{2}+1$, and there are infinitely many $n$'s when the hypercube can be covered with $n-\log_{2}(n)+1$ skewed hyperplanes. The…

For bipartite graphs $G$ and $H$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(G, H)$ of $G$ and $H$ is the smallest integer $n$, such that every red-blue coloring of $K_{m,n}$ results in a red $G$ or a blue $H$.…

After extending the classic notion of a tight Heffter array H$(m,n)$ to any group of order $2mn+1$, we give direct constructions for elementary abelian tight Heffter arrays, hence in particular for prime tight Heffter arrays. If $q=2mn+1$…

This paper gives a method to find all imaginary multiquadratic fields of class number dividing $2^{m},$ provided the list of all imaginary quadratic fields of class number dividing $2^{m+1}$ is known. We give a bound on the degree of such…

We start with the Lorentz algebra $ L=o_{R}(1,3)$ over the reals and find a suitable basis $B$ relative to which the structure constants are integers. Thus we consider the $Z$-algebra $L_{Z}$ which is free as a $Z$-module and its $Z$-basis…

Let $F$ be a totally real number field of class number one, and let $K$ be a CM-field with $F$ as its maximal real subfield. For each positive integer $N$, we construct a class group of certain binary quadratic forms over $F$ which is…

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$…

The resolvent degree $\textrm{rd}_{\mathbb{C}}(n)$ is the smallest integer $d$ such that a root of the general polynomial $$f(x) = x^n + a_1 x^{n-1} + \ldots + a_n$$ can be expressed as a composition of algebraic functions in at most $d$…

Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$…

This note aims at finding explicit and efficient generation of ideals in subalgebras $R$ of the polynomial ring $S=k[t]$ ($k$ a field) such that $t^{c_0}S \subseteq R$ for some integer $c_0 > 0$. The class of these subalgebras which we call…