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

Let f(x) = x(4 )+ ax(3) + d is an element of & Zopf;[x], where ad not equal 0. Let C-n denote the cyclic group of order n, D-4 the dihedral group of order 8, and A(4) 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 & Qopf;. In particular, we show that G = D4 if and only if (a,d) = (+/- 2, 2), and that G is not an element of {C-4,C-2 x C-2}. Furthermore, we prove that f(x) is monogenic with G = A(4) if and only if a = 4k and d = 27k(4) + 1, where k not equal 0 is an integer such that 27k(4) + 1 is squarefree. This paper extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups.