Criteria for the Irreducibility of Polynomials

Irreducibility Criteria

“…In this paper, new irreducibility criteria for a polynomial are introduced which are based on the functional values of its argument. Similar types of criteria have been obtained by Schur [1], Polya [2], Dorwart and Ore [3], Weisner [4], Brauer and Ehrlich [5] and Schultz [6]. In addition, Seres has validated a number of irreducibility conjectures of Schur; [7] contains a representative result.…”

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

“…By (10) we deduce that the leading coefficient of F , regarded as a polynomial in Y with coefficients…”

“…The first criterion of this kind was suggested by Schur [21], who raised the question of the irreducibility of the polynomials of the form (X − a 1 ) · · · (X − a n ) ± 1. For a unifying approach of the irreducibility criteria for polynomials of this type, we refer the interested reader to [10], [11] and [12]. …”

Polynomial products modulo primes and applications

“…We note that the previous argument is standard while working over Q, see for example, [7,Theorem 8] Proof. Let Q(α 1 ) denote the splitting field of P .…”