On monogenity of certain pure number fields of degrees $2^r\cdot 3^k\cdot 7^s$

Hamid Ben Yakkou; Jalal Didi

Displaying similar documents to “On monogenity of certain pure number fields of degrees $2^{r} \cdot 3^{k} \cdot 7^{s}$ ”

Variations on a question concerning the degrees of divisors of $x^{n} - 1$

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial $x^{n} - 1$ : “How often does $x^{n} - 1$ have a divisor of every degree between $1$ and $n$ ?” In a previous paper, we considered the situation when $x^{n} - 1$ is factored in $ℤ [x]$ . In this paper, we replace $ℤ [x]$ with $𝔽_{p} [x]$ , where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$ .

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

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We consider the equation $- y^{'} (x) + q (x) y (x - ϕ (x)) = f (x), x \in ℝ,$ where $ϕ$ and $q$ ( $q \geq 1$ ) are positive continuous functions for all $x \in ℝ$ and $f \in C (ℝ)$ . By a solution of the equation we mean any function $y$ , continuously differentiable everywhere in $ℝ$ , which satisfies the equation for all $x \in ℝ$ . We show that under certain additional conditions on the functions $ϕ$ and $q$ , the above equation has a unique solution $y$ , satisfying the inequality $∥ y^{'} ∥_{C (ℝ)} + {∥ q y ∥}_{C (ℝ)} \leq c {∥ f ∥}_{C (ℝ)},$ where the constant $c \in (0, \infty)$ does not depend on the choice of $f$ .

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a group and $ω (G)$ be the set of element orders of $G$ . Let $k \in ω (G)$ and $m_{k} (G)$ be the number of elements of order $k$ in $G$ . Let nse $(G) = {m_{k} (G) : k \in ω (G)}$ . Assume $r$ is a prime number and let $G$ be a group such that nse $(G) =$ nse $(S_{r})$ , where $S_{r}$ is the symmetric group of degree $r$ . In this paper we prove that $G ≅ S_{r}$ , if $r$ divides the order of $G$ and $r^{2}$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

On the unit group of a semisimple group algebra $𝔽_{q} S L (2, ℤ_{5})$

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of $𝔽_{q} S L (2, ℤ_{5})$ , where $𝔽_{q}$ is a finite field with $q = p^{k}$ elements for prime $p > 5,$ and $S L (2, ℤ_{5})$ denotes the special linear group of $2 \times 2$ matrices having determinant $1$ over the cyclic group $ℤ_{5}$ .

On the distribution of $(k, r)$ -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number $n$ is said to be a $(k, r)$ -integer if $n = a^{k} b$ , where $k > r > 1$ and $b$ is not divisible by the $r$ th power of any prime. We study the distribution of such $(k, r)$ -integers in the Piatetski-Shapiro sequence ${⌊ n^{c} ⌋}$ with $c > 1$ . As a corollary, we also obtain similar results for semi- $r$ -free integers.

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n \geq 3$ , let $u_{1}, ..., u_{n}$ be pairwise coprime integers $\geq 2$ , $𝒟$ a family of nonempty proper subsets of ${1, ..., n}$ with “enough” elements, and $ε$ a function $𝒟 \to {\pm 1}$ . Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_{i} - ε (I)$ for some $I \in 𝒟$ , but it does not divide $u_{1} \dots u_{n}$ ? We answer this question in the positive when the $u_{i}$ are prime powers and $ε$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if $ε_{0} \in {\pm 1}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

A new characterization for the simple group $PSL (2, p^{2})$ by order and some character degrees

Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi, Zahra Momen (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $| PSL (2, p^{2}) |$ such that $G$ has an irreducible character of degree $p^{2}$ and we know that $G$ has no irreducible character $θ$ such that $2 p ∣ θ (1)$ , then $G$ is isomorphic to $PSL (2, p^{2})$ . As a consequence of our result we prove that $PSL (2, p^{2})$ is uniquely determined by the structure of its complex group algebra.

Displaying similar documents to “On monogenity of certain pure number fields of degrees 2 r · 3 k · 7 s ”

Displaying similar documents to “On monogenity of certain pure number fields of degrees $2^{r} \cdot 3^{k} \cdot 7^{s}$ ”