A Generalization of an Irreducibility Theorem of I. Schur

A generalization of a second irreducibility theorem of I. Schur

in which cases either f(x) is irreducible or f(x) is the product of two irreducible polynomials of equal degree. If |an| = n > 1, then for some choice of a1, . . . , an−1 ∈ Z and a0 = ±1, we have that f(x) is reducible. I. Schur (in [8]) obtained this result in the special case that an = ±1. Further results along the nature of Theorem 1 are also discussed in [6]. The purpose of this paper is to...

Generalizations of some irreducibility results by Schur

Theorems of Sylvester and Schur

An old theorem of Sylvester states that a product of k consecutive positive integers each exceeding k is divisible by a prime greater than k. We shall give a proof of this theorem and apply it to prove a result of Schur on the irreducibility of certain polynomials.

Hilbert’s Irreducibility Theorem and the Larger Sieve

We describe an explicit version of Hilbert’s irreducibility theorem using a generalization of Gallagher’s larger sieve. We give applications to the Galois theory of random polynomials, and to the images of the adelic representation associated to elliptic curves varying in rational families.

Generalization of Darbo's fixed point theorem and application

In this paper, an attempt is made to present an extension of Darbo's theorem, and its applicationto study the solvability of a functional integral equation of Volterra type.