New insights into the interstellar medium of the dwarf galaxy IC 10:
  connection between magnetic fields, the radio--infrared correlation and star
  formation
  We present the highest sensitivity and angular resolution study at 0.32 GHz
of the dwarf irregular galaxy IC\,10, observed using the Giant Metrewave Radio
Telescope, probing $\sim45$ pc spatial scales. We find the galaxy-averaged
radio continuum spectrum to be relatively flat, with a spectral index $\alpha =
-0.34\pm0.01$ ($S_\nu \propto \nu^\alpha$), mainly due to a high contribution
from free--free emission. At 0.32 GHz, some of the H{\sc ii} regions show
evidence of free--free absorption as they become optically thick below
$\sim0.41$ GHz with corresponding free electron densities of $\sim11-22~\rm
cm^{-3}$. After removing the free--free emission, we studied the
radio--infrared relations on 55, 110 and 165 pc spatial scales. We find that on
all scales the non-thermal emission at 0.32 and 6.2 GHz correlates better with
far-infrared (FIR) emission at $70\,\mu$m than mid-infrared emission at
$24\,\mu$m. The dispersion of the radio--FIR relation arises due to variations
in both magnetic field and dust temperature, and decreases systematically with
increasing spatial scale. The effect of cosmic ray transport is negligible as
cosmic ray electrons were only injected $\lesssim5$ Myr ago. The average
magnetic field strength ($B$) of $12~\mu$G in the disc is comparable to that of
large star-forming galaxies. The local magnetic field is strongly correlated
with local star formation rate ($\mathrm{SFR}$) as $B \propto
\mathrm{SFR}^{0.35\pm0.03}$, indicating a star-burst driven fluctuation dynamo
to be efficient ($\sim10$ per cent) in amplifying the field in IC\,10. The high
spatial resolution observations presented here suggest that the high efficiency
of magnetic field amplification and strong coupling with SFR likely sets up the
radio--FIR correlation in cosmologically young galaxies.
