Related papers: Monogenic cyclotomic compositions

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$…

Let p and r be two primes and n, m be two distinct divisors of pr. Consider the n-th and m-th cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of one of them modulo the other one.…

For odd prime numbers $p < q$, let $\Phi_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $\Phi_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder…

Let $K$ be a global field and $n > 1$ an integer. We show $n$ is composite if and only if there is an irreducible polynomial $f(x) \in K[x]$ of degree $n$ which is reducible $q$-adically for all the primes $q$ of $K$.

Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more…

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5})…

We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the…

Given two cyclotomic polynomials $\Phi_n(x)$ and $\Phi_m(x)$, $n\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)\Phi_n (x)+b(x)\Phi_m(x),$$ with a(x) and b(x) integer polynomials.

We study the explicit factorization of $2^n r$-th cyclotomic polynomials over finite field $\mathbb{F}_q$ where $q, r$ are odd with $(r, q) =1$. We show that all irreducible factors of $2^n r$-th cyclotomic polynomials can be obtained…

We prove that the ideal used in recent works to categorify the cyclotomic integers is generated by a cyclotomic polynomial. Moreover, we publish a proof by T. Ekedahl that the $q$-binomial relations used in the tensor product of…

Cantor primes are primes p such that 1/p belongs to the middle-third Cantor set. One way to look at them is as containing the base-3 analogues of the famous Mersenne primes, which encompass all base-2 repunit primes, i.e., primes consisting…

Let $K=\mathbb{Q}(\alpha)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f(x)=x^{12}-m$, with $m\neq 1$ is a square free rational integer. In this paper, we prove that if $m \equiv 2$ or $3$ (mod…

In this paper we consider the polynomial sequence $(P_{n}^{\alpha,q}(x))$ that is orthogonal on $[-1,1]$ with respect to the weight function $x^{2q+1}(1-x^{2})^{\alpha}(1-x), \alpha>-1, q\in \mathbb N$; we obtain the coefficients of the…

Two naturally occurring matroids representable over Q are shown to be dual: the {\it cyclotomic matroid} $\mu_n$ represented by the $n^{th}$ roots of unity $1,\zeta,\zeta^2,...,\zeta^{n-1}$ inside the cyclotomic extension $Q(\zeta)$, and a…

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 prove that arithmetic is interpretable in any indecomposable polynomial ring (in any set of variables), and in addition we provide an alternative uniform proof of undecidability for all members in this class of rings.

For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal Z(R) of R[x], which is the set of polynomials that vanish on R, and the ideal Z(m), the polynomials that vanish on…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv…

It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to…