Related papers: Monogenic cyclotomic compositions

Let $f(t_1, \ldots, t_r, X)\in \mathbb{Z}[t_1, \ldots, t_r,X]$ be irreducible and let $a_1, \ldots, a_r\in \mathbb{Z} \smallsetminus \{0,\pm 1\}$. Under a necessary ramification assumption on $f$, and conditionally on the Generalized…

The scaled inverse of a nonzero element $a(x)\in \mathbb{Z}[x]/f(x)$, where $f(x)$ is an irreducible polynomial over $\mathbb{Z}$, is the element $b(x)\in \mathbb{Z}[x]/f(x)$ such that $a(x)b(x)=c \pmod{f(x)}$ for the smallest possible…

We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^n q$, where $p$, $q$ are distinct primes and $n\geq1$. This means that a subset $E$ of such a group $G$ tiles the group by translation…

Let l>=1 be an arbitrary odd integer and p,q and r primes. We show that there exist infinitely many ternary cyclotomic polynomials \Phi_{pqr}(x) with l^2+3l+5<= p<q<r such that the set of coefficients of each of them consists of the p…

Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a complex root $\alpha$ a monic irreducible polynomial $ F(x) = x^{p^r} -m$, with $ m \neq 1 $ is a square free rational integer, $p$ is a rational prime integer, and $r$ is…

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$,…

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas…

We investigate monogenicity and prime splitting in extensions generated by roots of iterated quadratic polynomials. Let $f(x)\in\mathbb{Z}[x]$ be an irreducible, monic, quadratic polynomial, and write $f^n(x)$ for the $n^{\text{th}}$…

Let $f(x)\in {\mathbb Z}[x]$ be monic of degree $N\ge 2$. Suppose that $f(x)$ is monogenic, and that $f(x)$ is the characteristic polynomial of the $N$th order linear recurrence sequence $\Upsilon_f:=(U_n)_{n\ge 0}$ with initial conditions…

Let $\Psi_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $\Psi_n(x)=(x^n-1)/\Phi_n(x)$, with $\Phi_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $\Psi_n(x)$…

For the $n$th cyclotomic polynomial $\Phi_n$, let $A(n)$ denote the greatest absolute value of its coefficients, its height, and let $D(n)$ denote the difference between its largest and smallest coefficients, its diameter. We show that for…

Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is doubly transitive simple non-abelian group. Let p be an odd prime, Z[\zeta_p] the ring of integers in the p-th…

The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N…

We show that the higher Pythagoras numbers for the polynomial ring are infinite $p_{2s}(K[x_1,x_2,\dots,x_n])=\infty$ provided that $K$ is a formally real field, $n\geq2$ and $s\geq 1$. This almost fully solves an old question \cite[Problem…

Let $k\geq 2$ be a square-free integer. We prove that the number of square-free integers $m\in [1,N]$ such that $(k,m)=1$ and $\mathbb{Q}(\sqrt[3]{k^2m})$ is monogenic is $\gg N^{1/3}$ and $\ll N/(\log N)^{1/3-\epsilon}$ for any…

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*}…

Let $q = p^s$ be a power of a prime number $p$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over $\mathbb{F}_q$ where both $r…

Let $c$ be a fixed integer such that $c \in \{0,2\}.$ Let $n$ be a positive integer such that either $n\geq 2$ or $2n+1 \neq 3^u$ for any integer $u\geq 2$ according as $c = 0$ or not. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic…

Let $\mathbb R_t[\theta]$ be the ring generated over $\mathbb R$ by $\cos\theta$ and $\sin\theta$, and $\mathbb R_t(\theta)$ be its quotient field. In this paper we study the ways in which an element p of $\mathbb R_t[\theta]$ can be…

The ring of integer-valued polynomials over a given subset $S$ of $\Z$ (or $ \mathrm{Int}(S,\Z ))$ is defined as the set of polynomials in $\Q[x]$ which maps $S$ to $\Z$. In factorization theory, it is crucial to check the irreducibility of…