Related papers: Simplest cubic fields with small class number

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive…

Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and…

Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$,…

Given $d\in\mathbb{N}$, let $\alpha(d)$ be the largest real number such that every abstract simplicial complex $\mathcal{S}$ with $0<\vert\mathcal{S}\vert\leq\alpha(d)\vert V(\mathcal{S})\vert$ has a vertex of degree at most $d$. We extend…

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs…

Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $\alpha$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=Q(\alpha)$ are called simplest sextic fields and…

We call an odd positive integer $n$ a $\textit{Descartes number}$ if there exist positive integers $k,m$ such that $n = km$ and \begin{equation} \sigma(k)(m+1) = 2km \end{equation} Currently, $\mathcal{D} = 3^{2}7^{2}11^{2}13^{2}22021$ is…

Let $m$ be a rational integer with $m \neq 0, \pm 1$, and consider the pure number field $K = \mathbb{Q}(\sqrt[n]{m})$ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus exclusively on the case where $m$ is…

We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even…

A positive integer $m$ is called a Hall number if any finite group of order precisely divisible by $m$ has a Hall subgroup of order $m$. We prove that, except for the obvious examples, the three integers $12$, $24$ and $60$ are the only…

For given simple graphs $H_1,H_2,\dots,H_c$, the multicolor Ramsey number $R(H_1,H_2,\dots,H_c)$ is defined as the smallest positive integer $n$ such that for an arbitrary edge-decomposition $\{G_i\}^c_{i=1}$ of the complete graph $K_n$, at…

It is proved that c = 689347 = 31*37*601 is the smallest conductor of a cyclic cubic number field K whose maximal unramified pro-3-extension E = F(3,infinity,K) possesses an automorphism group G = Gal(E/K) of order 6561 with coinciding…

We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one…

We develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the ``algebraic niceness'' phenomenon in $\mathbb{Z}_m$. We show that every integer $m = pq = 2^t -1$, where $p$,…

Let $d\ge4$ and $c\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\le d\le 36$), the smallest prime $p\equiv c\pmod d$ with $p\ge(2dn-c)/(d-1)$ is the…

Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and…

Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\dots,i_m\in\{\pm 1\}$ such that the product…

Let $L_n$ be a simplest cubic field with Galois group $G=\rm{Gal} (L_n/\mathbb Q)$. The associated order is denoted as ${\cal A}_{L_n/\mathbb Q}:= \{ x\in {\mathbb Q} [G] \, |\, x \cdot \cal{O}_{L_n} \subset {\cal O}_{L_n } \}$, where…

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and…