  It is shown here that the Exact Exchange (EE) formalism provides a natural
and rigorous approach for a Density Functional Theory (DFT) of the Integer
Quantum Hall Effect (IQHE). Application of a novel EE method to a quasi
two-dimensional electron gas (q2DEG) subjected to a perpendicular magnetic
field leads to the following main findings. \textit{i)} the microscopic
exchange energy functional of the IQHE has been obtained, whose main feature
being that it minimizes with a discontinuous derivative at every integer
filling factor $\nu$; \textit{ii)} an analytical solution is found for the
magnetic-field dependent EE potential, in the one-subband regime; \textit{iii)}
as a consequence of \textit{i)}, the EE potential display sharp discontinuities
at every integer $\nu$; and \textit{iv)} the widely used Local Spin Density
Approximation (LSDA) is strongly violated for filling factors close to integer
values.
