We present a catalog of Fe, Mg, Si, Ca, and Ti abundances for 2961 red giant stars that are likely members of eight dwarf satellite galaxies of the Milky Way (MW): Sculptor, Fornax, Leo I, Sextans, Leo II, Canes Venatici I, Ursa Minor, and Draco. For the purposes of validating our measurements, we also observed 445 red giants in MW globular clusters and 21 field red giants in the MW halo. The measurements are based on Keck/DEIMOS medium-resolution spectroscopy combined with spectral synthesis. We estimate uncertainties in [Fe/H] by quantifying the dispersion of [Fe/H] measurements in a sample of stars in monometallic globular clusters. We estimate uncertainties in Mg, Si, Ca, and Ti abundances by comparing our medium-resolution spectroscopic measurements to high-resolution spectroscopic abundances of the same stars. For this purpose, our DEIMOS sample included 132 red giants with published high-resolution spectroscopy in globular clusters, the MW halo field, and dwarf galaxies. The standard deviations of the differences in [Fe/H] and [alpha/Fe] (the average of [Mg/Fe], [Si/Fe], [Ca/Fe], and [Ti/Fe]) between the two samples is 0.15 and 0.16, respectively. This catalog represents the largest sample of multi-element abundances in dwarf galaxies to date. The next papers in this series draw conclusions on the chemical evolution, gas dynamics, and star formation histories from the catalog presented here. The wide range of dwarf galaxy luminosity reveals the dependence of dwarf galaxy chemical evolution on galaxy stellar mass.
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