  QED is studied at low temperature ($T\ll m$, where $m$ is the electron mass)
and zero chemical potential. By integrating out the electron field and the
nonzero bosonic Matsubara modes, we construct an effective three-dimensional
field theory that is valid at distances $R\gg1/T$. As applications, we
reproduce the ring-improved free energy and calculate the Debye mass to order
$e^5$.
