Related papers: Certifying Irreducibility in Z[x]

An algorithm for factoring polynomials over finite fields is given by Berlekamp in 1967. The main tool was the matrix Q corresponding to each polynomial. This paper studies the degrees of polynomials over binary field that associated with…

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1,…

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…

For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R = \mathbb{Z}/n$, extending a result of L. Carlitz. For…

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…

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related…

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…

We study the class of univariate polynomials $\beta_k(X)$, introduced by Carlitz, with coefficients in the algebraic function field $\mathbb F_q(t)$ over the finite field $\mathbb F_q$ with $q$ elements. It is implicit in the work of…

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…

Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences…

H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

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.

The Schinzel hypothesis claims (but it seems hopeless to prove) that any irreducible Q[x] polynomial without a constant factor assumes infinitely many prime values at integer places. On the other hand, it is easy to see that a reducible…

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…

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…

Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into…

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

We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property,…

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.