Related papers: A Note on Cyclotomic Polynomials

We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.

This paper investigates whether or not polynomials that are irreducible over $\mathbb{Q}$ and $\mathbb{Z}$ can remain irreducible under substitution by all quadratic polynomials. It answers this question in the negative in the degree 2 and…

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…

For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in…

In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.

We give a criterion for a quasi-ordinary polynomial to be irreducible. The criterion is based on the notion of approximate roots and that of generalized Newton polygons.

In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…

We present a theorem about irreducibility of a polynomial that is the resultant of two others polynomials. The proof of this fact is based on the field theory. We also consider the converse theorem and some examples.

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of…

We prove an irreducibility criterion for polynomials with power series coefficients generalizing previous known results concerning quasi-ordinary polynomials.

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

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…

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

We provide an alternative exposition of a result due to Schinzel. Fix an integer $k \ge 2$. For almost all choices of positive integers $n_{1} < \cdots < n_{k}$, we show that the polynomial $F(x) = 1 + x^{n_{1}} + \cdots + x^{n_{k}}$,…

In this article, we give two different sufficient conditions for the irreducibility of a polynomial of more than one variable, over the field of complex numbers, that can be written as a sum of two polynomials which depend on mutually…

We present several approaches on finding necessary and sufficient conditions on $n$ so that $\Phi_k(x^n)$ is irreducible where $\Phi_k$ is the $k$-th cyclotomic polynomial.

An irreducible polynomial over $\Bbb F_q$ is said to be normal over $\Bbb F_q$ if its roots are linearly independent over $\Bbb F_q$. We show that there is a polynomial $h_n(X_1,\dots,X_n)\in\Bbb Z[X_1,\dots,X_n]$, independent of $q$, such…