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We compute the Galois group of the splitting field $F$ of any irreducible andseparable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field withcharacteristic different from two. The proofs require to distinguish betweentwo cases: whether or not the cubic roots of unity belong to $K$. We also givea 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 acomplete list of all the possible subfields of $F$.

An elementary computation of the Galois groups of symmetric sextic trinomials