[Abridged] We study the evolution of the radio spectral index and far-infrared/radio correlation (FRC) across the star-formation rate-stellar masse (i.e. SFR-M*) plane up to z 2. We start from a M*-selected sample of galaxies with reliable SFR and redshift estimates. We then grid the SFR-M* plane in several redshift ranges and measure the infrared luminosity, radio luminosity, radio spectral index, and ultimately the FRC index (i.e. qFIR) of each SFR-M*-z bin. The infrared luminosities of our SFR-M*-z bins are estimated using their stacked far-infrared flux densities inferred from observations obtained with Herschel. Their radio luminosities and radio spectral indices (i.e. alpha, where Snu nu^-alpha) are estimated using their stacked 1.4GHz and 610MHz flux densities from the VLA and GMRT, respectively. Our far-infrared and radio observations include the most widely studied blank extragalactic fields -GOODS-N/S, ECDFS, and COSMOS- covering a sky area of 2deg^2. Using this methodology, we constrain the radio spectral index and FRC index of star-forming galaxies with M*>10^10Msun and 0<z<2.3. We find that alpha^1.4GHz_610MHz does not evolve significantly with redshift or with the distance of a galaxy with respect to the main sequence (MS) of the SFR-M* plane (i.e. Delta_log(SSFR)_MS=log[SSFR(galaxy)/SSFR_MS(M*,z)]). Instead, star-forming galaxies have a radio spectral index consistent with a canonical value of 0.8, which suggests that their radio spectra are dominated by non-thermal optically thin synchrotron emission. We find that qFIR displays a moderate but statistically significant redshift evolution as qFIR(z)=(2.35+/-0.08)*(1+z)^(-0.12+/-0.04), consistent with some previous literature. Finally, we find no significant correlation between qFIR and Delta_log(SSFR)_MS, though a weak positive trend, as observed in one of our redshift bins, cannot be firmly ruled out using our dataset.
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