We investigate the correlation between far-infrared (FIR) and radio luminosities in distant galaxies, a lynchpin of modern astronomy. We use data from the Balloon-borne Large Aperture Submillimetre Telescope (BLAST), Spitzer, the Large Apex BOlometer CamerA (LABOCA), the Very Large Array (VLA) and the Giant Metre-wave Radio Telescope (GMRT) in the Extended Chandra Deep Field South (ECDFS). For a catalogue of BLAST 250-micron-selected galaxies, we re-measure the 70--870-micron flux densities at the positions of their most likely 24-micron counterparts, which have a median [interquartile] redshift of 0.74 [0.25, 1.57]. From these, we determine the monochromatic flux density ratio, q_250 = log_10 (S_250micron / S_1400MHz), and the bolometric equivalent, q_IR. At z ~= 0.6, where our 250-micron filter probes rest-frame 160-micron emission, we find no evolution relative to q_160 for local galaxies. We also stack the FIR and submm images at the positions of 24-micron- and radio-selected galaxies. The difference between q_IR seen for 250-micron- and radio-selected galaxies suggests star formation provides most of the IR luminosity in ~< 100-uJy radio galaxies, but rather less for those in the mJy regime. For the 24-micron sample, the radio spectral index is constant across 0 < z < 3, but q_IR exhibits tentative evidence of a steady decline such that q_IR is proportional to (1+z)^(-0.15 +/- 0.03) - significant evolution, spanning the epoch of galaxy formation, with major implications for techniques that rely on the FIR/radio correlation. We compare with model predictions and speculate that we may be seeing the increase in radio activity that gives rise to the radio background.
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