# An elementary computation of the Galois groups of symmetric sextic trinomials

## TL;DR

This paper provides a straightforward method for computing the Galois groups of certain sextic polynomials, with explicit criteria and classifications based on the field properties and polynomial irreducibility.

## Contribution

It introduces an elementary approach to determine Galois groups of symmetric sextic trinomials, including criteria for irreducibility over finite fields and a classification of subfields.

## Key findings

- Explicit Galois group computations for $x^6+ax^3+b$
- Irreducibility criteria over finite fields
- Complete classification of subfields of the splitting field

## Abstract

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases: whether or not the cubic roots of unity belong to $K$. We also give a criterion to determine whether a polynomial as $f(x)$ is irreducible, when $F$ is a finite field. Moreover, at the end of the paper we also give a complete list of all the possible subfields of $F$.