Monogenic trinomials of the form x⁴+ax³+d and their Galois groups

Let f(x)=x⁴+ax³+d∈ Z[x], where ad≠ 0. Let Cₙ denote the cyclic group of order n, D₄ the dihedral group of order 8, and A₄ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only a and d to determine the Galois group G of $f(x)$ over Q. In particular, we show that G=D₄ if and only if (a,d)=(± 2,2), and that G ∈ ₄,C₂× C₂\. Furthermore, we prove that $f(x)$ is monogenic with G=A₄ if and only if $a=4k$ and d=27k⁴+1, where k≠ 0 is an integer such that 27k⁴+1 is squarefree. This article extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups.