Irreducibility and Galois groups of truncated binomial polynomials

Abstract

For positive integers [Formula: see text], let [Formula: see text] be the truncated binomial expansion of [Formula: see text] consisting of all terms of degree [Formula: see text] It is conjectured that for [Formula: see text], the polynomial [Formula: see text] is irreducible. We confirm this conjecture when [Formula: see text] Also we show for any [Formula: see text] and [Formula: see text], the polynomial [Formula: see text] is irreducible when [Formula: see text] Under the explicit abc-conjecture, for a fixed [Formula: see text], we give an explicit [Formula: see text] depending only on [Formula: see text] such that [Formula: see text], the polynomial [Formula: see text] is irreducible. Further [Formula: see text], the Galois group associated to [Formula: see text] is the symmetric group [Formula: see text]