An elementary computation of the Galois groups of symmetric sextic trinomials

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 $F$.