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In this paper, the induced Chern-Simons like action for a system of fermionsinteracting with a gauge field in (3 +1) dimensions is calculated in thepresence of a background field that breaks the Lorentz and CPT symmetries. Themain result of this work corresponds to the obtation of a usualChern-Simons-type action in four-dimensional spacetime resulting from theaddition to the conventional QED Lagrangian of a term with a background fieldthat breaks the Lorentz symmetry of the Lagrangian gauge theory. It is pointedout that the proportionality constant in that term depends heavily on theregularization method used in the computations. As the articles on this subjectin the literature are very difficult to read, here calculations are performedclearly and objectively.

Induced Chern-Simons Like Action by Lorentz Symmetry Breaking in (3+1)D QED