Irreducibility of polynomials with a large gap

Abstract

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $\mathbb Q$, where $c, d \in \mathbb Z[x]$ are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that $x^N - k x^2 + 1$ is irreducible for all $N \ge 5$ and $k \in \{3, 4, \ldots, 24\} \setminus \{9, 16\}$. We also give a complete description of the factorization of polynomials of the form $x^N + k x^{N-1} \pm (l x + 1)$ with $k, l \in \mathbb Z$, $k \neq l$.