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arXiv · 1803.02754

On the Galois group over Q of a truncated binomial expansion

Michael Filaseta·Richard Moy
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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 $O(\log N)$.

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Michael Filaseta, Richard Moy. 2018-03-07. On the Galois group over Q of a truncated binomial expansion. https://arxiv.org/abs/1803.02754

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