Irreducibility criteria of Schur-type and Pólya-type

On some norm inequalities and discriminant inequalities in CM-fields

“…The above norm inequalities were applied in Győry and Lovász [7], Győry [11], and Aubry and Poulakis [1] to diophantine equations, and in Győry [8], [9], [10], [12], [13], Schinzel [24], and Győry, Hajdu and Tijdeman [14] to irreducible polynomials.…”

“…We repeat this procedure for i Im α and Re α as well. If there exists a ϕ j such that (Re α)ϕ j = Re α and (i Im α)ϕ j = i Im α, then it follows from (12), (13) and (14) that It remains the case when, for each j, (Re α)ϕ j = Re α or (i Im α)ϕ j = i Im α. But the number of Q-isomorphisms which leave Re α resp.…”

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

“…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

“…Throughout the years, arithmetic properties of the function F P (n) attracted considerable attention of several authors. Perhaps the first occurrence of this object in the literature dates back to Chebyshev, who considered the case P 0 (n) = n 2 + 1 and showed that the largest prime factor of F P 0 (n) is ≫ n. See also [12], [13], [15] for refinements of the latter result. Another direction in this investigation stems from equations of the form F P (n) = m k (1) where m, n ∈ N, k ≥ 2.…”