On the Galois group over Q of a truncated binomial expansion

TL;DR

This paper investigates the Galois groups of truncated binomial expansion polynomials over Q, proving they are typically symmetric groups for large n, with specific exceptions for r=6.

Contribution

It establishes that for fixed r ≠ 6, the Galois group is generically the symmetric group S_r for large n, and quantifies the rarity of exceptions when r=6.

Findings

Galois group is S_r for fixed r ≠ 6 and large n

Number of exceptions for r=6 grows at most logarithmically with N

Polynomials are generally irreducible for large n

Abstract

For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $\le r$ where $1 \le r \le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. We show here that for a fixed positive integer $r \ne 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. For $r = 6$, we show the number of exceptional $n \le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(\log N)$.