Computing Galois groups of certain families of polynomials
Volume 185 / 2018
Acta Arithmetica 185 (2018), 357-365
MSC: Primary 11R32; Secondary 11R29, 11S15.
DOI: 10.4064/aa171014-26-2
Published online: 8 October 2018
Abstract
There are several results on computing the Galois groups of different families of trinomials with integer coefficients. However, adding even one more term to a trinomial makes this computation very complicated due to some problems including the complexity of finding a formula for the discriminant. We show that the Galois group of certain irreducible quadrinomials of the form $f(x)=x^{n}+ax^{n-1}+bx^{n-2}+c\in \mathbb{Z}[x]$ is the full symmetric group $S_n$, and furthermore we generalize the results to a certain family of polynomials with an arbitrary number of terms.
