Reducibility of translates of Dickson polynomials

Abstract:

Let $K$ be a field and $a,b\in K$. The Dickson polynomial $D_{n}(x,a)$ is characterized by the equation $D_{n}(x+(a/x),a)=x^{n}+ (a/x)^{n}$. We prove that $D_{n}(x,a)+b\in K[x]$ is reducible if and only if there is a prime $p|n$ such that $b=-D_{p}(c,a^{n/p})$ for some $c\in K$, or $n=4k$ and $b=4c^{4}-8a^{k}c^{2}+2a^{2k}$ for some $c\in K$. This result generalizes the well-known reducibility criterion for binomials; and it provides a reducibility criterion for $T_{n}(x)+c$ where $T_{n}(x)$ denotes the Chebyshev polynomial of degree $n$.