Year: 2026
Type: preprint
Abstract: We calculate the Galois group of $f = x^n + ax^k + b \in \mathbb{Z}\left[ x \right]$ over $\mathbb{Q}$ for all $1 \leq \left| ab \right| \leq 3$, $n$ sufficiently large and $k$ coprime to $n$. In particular, we prove that the Galois group of $f$ is the maximal one that its factorization type allows. We give a criterion for the Galois group of a factor of a trinomial to be symmetric, with an eye towards future applications. To the best of our knowledge, the only previously known explicit example of polynomials of degree $n$ with bounded coefficients and Galois group $S_n$ is $x^n - x - 1$, while recent results show that the probability that a random polynomial of degree $n$ with reasonably bounded coefficients has Galois group $S_n$ tends to $1$ as $n$ tends to infinity. Using our calculation, one can construct many new such explicit examples.
Source: arXiv (Cornell University)
Author: Garmi, Hilel
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222944857