Variations on a theorem of Capelli

Abstract

Elementary irreducibility criteria are established for $f(x^p)$ where $f(x)\in \mathbb{Z}[x]$ is irreducible over $\mathbb{Q}$ and $p$ is a prime. For instance, our main criterion implies that if $f(x^p)$ is reducible over $\mathbb{Q}$, then $f(x)$ divides $f(x^p)$ modulo $p^2$. Among several applications, it is shown that if $f(x)$ has coefficients in $\{-1,1\}$, then $f(x^2)$ is irreducible over $\mathbb{Q}$ excluding a couple of obvious exceptions. As another application, it is proved that if $n>4$ and $a_1,a_2,\dots ,a_n$ are distinct integers, then for $𝜀\in \{-1,1\}$, the polynomial $(x^2-a_1)(x^2-a_2)\cdots (x^2-a_n)+𝜀$ is irreducible over $\mathbb{Q}$ unless $n$ is odd and $𝜀=-1$. Some emphasis is given to the non-cyclotomic monic polynomials $f(x)$ with $f(0)\in \{-1,1\}$. In these cases, among other things, it is shown that if $p\gg (degf)logmax\{2,H(f)\}$, where $H(f)$ denotes the height of $f(x)$, then $f(x^p)$ is irreducible over $\mathbb{Q}$. Proofs of the irreducibility criteria rest upon a general result of Capelli concerning the factorization of $f(x^m)$.