  We present for the first time a detailed spectroscopic study of chemical
element abundances of metal-poor RR Lyrae stars in the Large and Small
Magellanic Cloud (LMC and SMC). Using the MagE echelle spectrograph at the 6.5m
Magellan telescopes, we obtain medium resolution (R ~ 2000 - 6000) spectra of
six RR Lyrae stars in the LMC and three RR Lyrae stars in the SMC. These stars
were chosen because their previously determined photometric metallicities were
among the lowest metallicities found for stars belonging to the old populations
in the Magellanic Clouds. We find the spectroscopic metallicities of these
stars to be as low as [Fe/H]_{spec} = -2.7dex, the lowest metallicity yet
measured for any star in the Magellanic Clouds. We confirm that for metal-poor
stars, the photometric metallicities from the Fourier decomposition of the
lightcurves are systematically too high compared to their spectroscopic
counterparts. However, for even more metal-poor stars below [Fe/H]_{phot} <
-2.8dex this trend is reversed and the spectroscopic metallicities are
systematically higher than the photometric estimates. We are able to determine
abundance ratios for ten chemical elements, which extend the abundance
measurements of chemical elements for RR Lyrae stars in the Clouds beyond
[Fe/H] for the first time. For the overall [alpha/Fe] ratio, we obtain an
overabundance of 0.36dex, which is in very good agreement with results from
metal-poor stars in the Milky Way halo as well as from the metal-poor tail in
dwarf spheroidal galaxies. Comparing the abundances with those of the stars in
the Milky Way halo we find that the abundance ratios of stars of both
populations are consistent with another. Therefore we conclude that from a
chemical point of view early contributions from Magellanic-type galaxies to the
formation of the Galactic halo as claimed in cosmological models are plausible.
