We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G= hbox {exp}({mathfrak {g}}) by showing that operator fields defined on suitable sub-manifolds of {mathfrak {g}}^* are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of L^1(G) as kernels of sets of irreducible representations of G.