  We present constraints on the spacetime variation of the fine-structure
constant $\alpha$ at redshifts $3<z<10$ using JWST emission-line galaxies. The
galaxy sample consists of 572 high-quality spectra with strong and narrow [O
III] $\lambda\lambda$4959,5007 doublet emission lines from 522 galaxies,
including 267 spectra at $z>5$. The [O III] doublet lines are arguably the best
emission lines to probe the variation in $\alpha$. We divide our sample into 5
subsamples based on redshift and calculate the relative variation
$\Delta\alpha/\alpha$ for the individual subsamples. The calculated
$\Delta\alpha/\alpha$ values are consistent with zero within $1\sigma$ at all
redshifts, suggesting no time variation in $\alpha$ above a level of $(1-2)
\times10^{-4}$ ($1\sigma$) in the past 13.2 billion years. When the whole
sample is combined, the constraint is improved to be $\Delta\alpha/\alpha =
(0.4\pm0.7) \times10^{-4}$. We further test the spatial variation in $\alpha$
using four subsamples of galaxies in four different directions on the sky. The
measured $\Delta\alpha/\alpha$ values are consistent with zero at a $1\sigma$
level of $\sim10^{-4}$. While the constraints in this work are not as stringent
as those from lower-redshift quasar absorption lines in previous studies, this
work uses an independent tracer and provides the first constraints on
$\Delta\alpha/\alpha$ at the highest redshifts. Our analyses also indicate that
the relative wavelength calibration of the JWST spectra is robust. With the
growing number of emission-line galaxies from JWST, we expect to achieve
stronger constraints in the future.
