  Spectra have been obtained with the Infrared Spectrograph (IRS) on the
Spitzer Space Telescope for 20 sources in the Lockman Hole field of the SWIRE
survey. The sample is divided between sources with indicators of an obscured
AGN, based primarily on X-ray detections of optically-faint sources, and
sources with indicators of a starburst, based on optical and near-infrared
spectral energy distributions (SEDs) which show a luminosity peak from stellar
photospheric emission. Ten of the 11 AGN sources have IRS spectra which show
silicate absorption or are power laws; only one AGN source shows PAH emission
features. All 9 of the sources showing starburst SEDs in the near-infrared show
PAH emission features in the IRS spectra. Redshifts are determined from the IRS
spectra for all 9 starbursts (1.0 < z < 1.9) and 8 of the 11 AGN (0.6 < z <
2.5). Classification as AGN because of an X-ray detection, the classification
as AGN or starburst derived from the photometric SED, and the IRS spectroscopic
classification as AGN (silicate absorption) or starburst (PAH emission) are all
consistent in 18 of 20 sources. The surface density for starbursts which are
most luminous in the mid-infrared is less than that for the most luminous AGN
within the redshift interval 1.7 < z < 1.9. This result implies that
mid-infrared source counts at high redshift are dominated by AGN for
f(24micron) > 1.0 mJy.
