  Over the past few years three dimensional Quantum Electrodynamics (QED$_3$)
has attracted a lot of attention, because it may be an effective theory for the
underdoped and non-superconducting region of the phase diagram of high $T_c$
cuprate compounds. We present results from lattice simulations of the
non-compact version of the theory in order to address the issue of whether
chiral symmetry is spontaneously broken when the number of fermion flavours
$N_f$ is less than a critical value $N_{fc}$. Our results provide strong
evidence that QED$_3$ is chirally symmetric for $N_f \geq 1.5$, implying that a
pseudogap phase separates the superconducting phase from the antiferromagnetic
phase.
