  The high-frequency radio sky has historically remained largely unexplored due
to the typical faintness of sources in this regime, and the modest survey speed
compared to observations at lower frequencies. However, high-frequency radio
surveys present an invaluable tracer of high-redshift star-formation, as they
directly target the faint radio free-free emission. We present deep continuum
observations at 34 GHz in the COSMOS and GOODS-North fields from the Karl G.
Jansky Very Large Array (VLA), as part of the COLDz survey. The deep COSMOS
mosaic spans $\sim10\text{arcmin}^2$ down to $\sigma=1.3\mu\text{Jy
beam}^{-1}$, while the wider GOODS-N observations cover $\sim50\text{arcmin}^2$
to $\sigma=5.3\mu\text{Jy beam}^{-1}$. We present the deepest 34 GHz radio
number counts to date, with five and thirteen continuum detections in COSMOS
and GOODS-N, respectively. Nine galaxies show 34 GHz continuum emission that is
originating from star-formation, although for two sources this is likely due to
thermal emission from dust. Utilizing deep ancillary radio data at 1.4, 3, 5
and 10 GHz, we decompose the spectra of the remaining seven star-forming
galaxies into their synchrotron and thermal free-free components, finding
typical thermal fractions and synchrotron spectral indices comparable to those
observed in local star-forming galaxies. Using calibrations from the
literature, we determine free-free star-formation rates (SFRs), and show that
these are in agreement with SFRs from spectral energy distribution fitting and
the far-infrared/radio correlation. Our observations place strong direct
constraints on the high-frequency radio emission in typical galaxies at
high-redshift, and provide some of the first insight in what is set to become a
key area of study with future radio facilities as the Square Kilometer Array
Phase 1 and next-generation VLA.
