Title: Do Globular Clusters Share the Chemical Enrichment History of their Host Galaxy? Abstract: The large majority of Milky Way Halo globular clusters (GCs) have super-solar alpha-element to iron abundance ratios and by that trace the ratio for Galactic field stars at comparable metallicity. The standard interpretation is that they formed quickly out of ''pristine'' material which did not have time to be polluted especially by SN Ia byproducts. In contrast, some unusual clusters in the outer Halo display extremely low [alpha/Fe] ratios. These clusters could have formed in an environment of very slow chemical enrichment, e.g., dwarf spheroidal galaxies (dSphs), where the GC compositions reflect the enrichment process of their galactic environment. Here, we present the first high-resolution spectroscopy for individual stars in the unusual GC H4 in the Fornax dSph. With [Fe/H]=-1.4, H4 is significantly more metal-rich than the remainder of Fornax'' GC population. We have recently shown that at this metallicity, field stars in Fornax have already experienced significant SNe Ia contributions and accordingly low [alpha/Fe] ratios. The key question is thus whether H4''s alpha-element ratios are similarly depleted in these elements, and by that follow the trend of the field. From our data, we obtain precise [alpha/Fe] abundances, metallicities and the age of the cluster. The results are interpreted with respect to the chemical evolution of field stars in Fornax from which we obtain important clues on the link between the chemical enrichment of GCs and their host galaxy and the origin of alpha-depleted outer Halo GCs.