Irreducibility criteria for compositions of polynomials with integer coefficients (Q515627)

The authors prove several results about the irreducibility of polynomials of the form \(F(G(x))\), where \(F\) is a quadratic irreducible polynomial and \(G\) is a polynomial of arbitrary degree. This is the case, for instance, for \(F(x)=ax^2+bx+c \in {\mathbb Z}[x]\), where \(a=pq\) with \(p\) prime and \(q \in {\mathbb Z}\) such that \(p\) does not divide \(cq\), and any non-constant polynomial \(G \in {\mathbb Z}[x]\) whose leading coefficient is not divisible by \(p\) (Theorem 1). Three other (more technical) theorems of this type are also proved.