Irreducibility of polynomials with a large gap

2018/03/28 by William Sawin, Mark Shusterman, Sawin, William +3
Mathematics · #11C08 #11R09 #12E05 #13P05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11C08 #msc:11R09 #msc:12E05 #msc:13P05

paper · pdf · doi:10.48550/arxiv.1803.10811

25 pages, 2 figures

arxiv created 2018/03/28 · arxiv updated 2018/03/30

Abstract

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of N > deg(c) + deg(d) such that the polynomial f_N(x) = x^N c(x^-1) + d(x) is irreducible over \mathbb Q, where c, d ∈ \mathbb Z[x] are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that x^N - k x² + 1 is irreducible for all N ≥ 5 and k ∈ \3, 4, …, 24\ ∖ \9, 16\. We also give a complete description of the factorization of polynomials of the form x^N + k x^N-1 ± (l x + 1) with k, l ∈ \mathbb Z, k ≠ l.

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