Irreducibility of polynomials with a large gap

TL;DR

This paper develops a practical algorithm to determine the irreducibility of a class of polynomials with large gaps, extending classical results and providing new irreducibility criteria for specific polynomial families.

Contribution

It generalizes Ljunggren's approach to create an algorithm for identifying irreducible polynomials with large gaps and applies it to characterize irreducibility in specific polynomial forms.

Findings

The polynomial x^N - k x^2 + 1 is irreducible for all N ≥ 5 and certain k.

Complete factorization descriptions for polynomials of the form x^N + k x^{N-1} ± (l x + 1).

An effective method to determine irreducibility of polynomials with large gaps.

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 $k, l \in \mathbb Z$, $k \neq l$.