  A self-consistent treatment of exchange and correlation interactions in a
quantum wire (QW) subject to a strong perpendicular magnetic field is presented
using a modified local-density approximation (MLDA). The influence of many-body
interactions on the spin-splitting between the two lowest Landau levels (LLs)
is calculated within the screened Hartree-Fock approximation (SHFA), for
filling factor \nu=1, and the strong spatial dependence of the screening
properties of electrons is taken into account. In comparison with the
Hartree-Fock result, the spatial behavior of the occupied LL in a QW is
strongly modified when correlations are included. Correlations caused by
screening at the edges strongly suppress the exchange splitting and smoothen
the energy dispersion at the edges. The theory accounts well for the
experimentally observed strong suppression of the spin-splitting pertinent to
the \nu=1 quantum Hall effect (QHE) state as well as the destruction of this
state in long, quasi-ballistic GaAlAs/GaAs QWs.
