On the Galois group over $\mathbb Q$ of a truncated binomial expansion
Volume 154 / 2018
Colloquium Mathematicum 154 (2018), 295-308
MSC: Primary 11R09, 12F10; Secondary 11R32, 12E05, 12E10.
DOI: 10.4064/cm7474-3-2018
Published online: 24 September 2018
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 $r \not =6$ 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 that the number of exceptional $n \le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(\log N)$.
