  Using an effective field theory approach, we address the effects on the gauge
couplings of one and two additional compact dimensions in the presence of a
constant background (gauge) field. Such background fields are a generic
presence in models with extra dimensions and can be employed for gauge symmetry
breaking mechanisms in the context of 4D N=1 supersymmetric models. The
structure of the ultraviolet (UV) and infrared (IR) divergences that the gauge
couplings develop in the presence of Wilson line vev's is investigated.
One-loop radiative corrections to the gauge couplings due to overlapping
effects of the compact dimensions and Wilson line vev's are computed for
generic 4D N=1 models. Values of Wilson lines vev's corresponding to points (in
the ``moduli'' space) of enhanced gauge symmetry cannot be smoothly reached
{\it perturbatively} from those corresponding to the broken phase. The one-loop
corrections are compared to their (heterotic) string counterpart in the ``field
theory'' limit alpha'->0 to show remarkably similar results when no massless
states are present in a Kaluza-Klein tower. An additional correction to the
gauge coupling exists in the effective field theory approach when for specific
Wilson lines vev's massless Kaluza-Klein states are present. This correction is
not recoverable by the limit alpha'->0 of the (infrared regularised) string
because the infrared regularisation limit and the limit alpha'->0 of the string
result do not commute.
