We present the results of spectroscopy using HYDRA on the WIYN 3.5m telescope of objects in the deep SWIRE radio field. The goal of the project was to determine spectroscopic redshifts for as many of the brighter objects in the field as possible, especially those detected in the radio and at 24um. These redshifts are primarily being used in studies of galaxy evolution and the connection of that evolution to active galactic nuclei and star formation. Redshifts measured for 365 individual objects are reported. The redshifts range from 0.03 to 2.5, mostly with z<0.9. The sources were selected to be within the WIYN HYDRA field of approximately 30' in radius from the center of the SWIRE deep field, 10:46:00, +59:01:00 (J2000). Optical sources for spectroscopic observation were selected from an r-band image of the field. A priority list of spectroscopic targets was established in the following order: 20cm detections, 24m detections, galaxies with r<20 and the balance made up of fainter galaxies in the field. We provide a table listing the galaxy positions, measured redshift and error, and note any emission lines that were visible in the spectrum. In practice, almost all the galaxies with r<19 were observed including all of the radio sources and most of the 24um sources with r<20 and a sample of radio sources which had fainter optical counterparts on the r-band image.