On an irreducibility theorem of I. Schur

A Generalization of an Irreducibility Theorem of I. Schur

is irreducible. Irreducibility here and throughout this paper refers to irreducibility over the rationals. Some condition, such as ja0j = janj = 1, on the integers aj is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial inZ[x] is irreducible (which is clearly not the case). In this paper, we will mainly be interested in relaxing the condi...

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...

On Hilbert’s Irreducibility Theorem

In this paper we obtain new quantitative forms of Hilbert’s Irreducibility Theorem. In particular, we show that if f(X,T1, . . . , Ts) is an irreducible polynomial with integer coefficients, having Galois group G over the function field Q(T1, . . . , Ts), and K is any subgroup of G, then there are at most Of,ε(H s−1+|G/K|+ε) specialisations t ∈ Zs with |t| ≤ H such that the resulting polynomial...

Generalizations of some irreducibility results by Schur

On a Theorem of Schur

We study the ramifications of Schur’s theorem that, if G is a group such that G/ZG is finite, then G′ is finite, if we restrict attention to nilpotent group. Here ZG is the center of G, and G′ is the commutator subgroup. We use localization methods and obtain relativized versions of the main theorems. 2000 Mathematics Subject Classification. 20B07, 20D15.