Related papers: Swan-like reducibility for Type I pentanomials ove…

In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…

For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\in \mathbb Z)$ are irreducible and their…

For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ of degree $k$, a family of polynomials $F_m(x)$ depending on $f(x)$ is constructed with the properties: (i) there is exactly one irreducible factor $\Phi_{d,f}(x)$…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

We give, in this paper, all bi-unitary perfect polynomials over the prime field $\mathbb{F}_2$, with at most four irreducible factors.

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

In this work, we completely characterize (i) permutation binomials of the form $x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb{F}_{2^n}[x], n = 2^st, a \in \mathbb{F}_{2^{2t}}^{*}$, and (ii) permutation trinomials of the form…

Permutation polynomials with few terms attracts researchers' interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree…

In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree $n$ over a finite field and their roots in the extension field in quasilinear where $N=n^2$…

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n = (q+1)/2$. For each $s \in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y): = y^n + (1-y)^n -s$, and we give a necessary and…

The famous irreducibility criteria of Sch\"onemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper we provide several irreducibility criteria of…

Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of…

Let $a,b,c$ be non-zero integers and $f(x)=ax^n+bx^m+c$ be a trinomial of degree $n$. We surveyed the irreducibility criteria of $f(x)$ over rational numbers.

Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational…

Using Newton polygons, a key factorization result for polynomials over discrete valuation domains is proved, which in particular yields new irreducibility criteria including a generalization of the classical irreducibility criterion of…

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…

This paper addresses the factorization of polynomials of the form $F(x) = f_{0}(x) + f_{1}(x) x^{n} + \cdots + f_{r-1}(x) x^{(r-1)n} + f_{r}(x) x^{rn}$ where $r$ is a fixed positive integer and the $f_{j}(x)$ are fixed polynomials in…

We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f.…

Let f\in \mathbb{Z}[x,y] be an irreducible homogeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.