  We present two compact derivations of the correct definition of the
Chern-Simons term in the topologically non trivial context of thermal $QED_3$.
One is based on a transgression descent from a D=4 background connection, the
other on embedding the abelian model in SU(2). The results agree with earlier
cohomology conclusions and can be also used to justify a recent simple
heuristic approach. The correction to the naive Chern-Simons term, and its
behavior under large gauge transformations are displayed.
