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

Let f (x) =x⁴+ax³+d Zx, 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 \C₄, 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.