Detailed observations of the Milky Way can be used to test predictions from simulations of disk formation and evolution, and they serve to complement large galaxy surveys at high redshift. The observed elemental abundances of old stars in the Milky Way disk, and how they vary with location in the Galaxy, provide powerful constraints on the chemical enrichment and assembly history of the disk. We present trends in [Fe/H] and [α/Fe] as a function of Galactocentric radius R and distance from the plane |Z| using main sequence turnoff stars observed by the Sloan Extension for Galactic Understanding and Exploration survey. We find that the thick disk of the Milky Way has no radial metallicity gradient and is chemically homogeneous, whether thick disk stars are identified by their location (i.e., far from the midplane) or by their chemistry (i.e., α-enhanced). Follow-up observations from the High Resolution Echelle Spectrometer at the W. M. Keck Observatory show that the α-, iron peak, and neutron capture elemental abundances of stars at |Z| ~ 0.5 kpc also resemble those in the solar neighborhood. In addition, we find that the high-α population has a short radial scale length. The observed chemical homogeneity is consistent with a cosmological origin for the α-enhanced thick disk, in which stars far from the midplane of the disk formed at early times during a chaotic period of high gas accretion when the disk was turbulent and clumpy. The observations can only be explained by radial mixing processes, such as disk heating during a minor merger or internal radial migration processes, if mixing was extremely efficient.